It seems that there are two distinct stages of damaging the soul. The mere act of murder is said to rip apart the soul. This is what Slughorn tells Tom Riddle in Chapter Twenty Three of Half-Blood Prince:

"By an act of evil — the supreme act of evil. By committing murder. Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: He would encase the torn portion —"

And this is what Dumbledore indicates to Snape in Chapter Thirty Three of Deathly Hallows:

“If you don’t mind dying,” said Snape roughly, “why not let Draco do it?”

“That boy’s soul is not yet so damaged,” said Dumbledore. “I would not have it ripped apart on my account.”

Muggles can presumably achieve the state of having a ripped apart soul, as they can commit murder just like wizards can. However, Dumbledore's statement in Chapter Thirty Five of Deathly Hallows about Harry being a "Horcrux" is very specific:

“You were the seventh Horcrux, Harry, the Horcrux he never meant to make. He had rendered his soul so unstable that it broke apart when he committed those acts of unspeakable evil, the murder of your parents, the attempted killing of a child. But what escaped from that room was even less than he knew. He left more than his body behind. He left part of himself latched to you, the would-be victim who had survived.

This implies that Voldemort had gone beyond the mere ripping apart that happens to anyone who murders. By creating previous Horcruxes, Voldemort had rendered his soul entirely unstable.

Hermione's statement in Chapter Six of Deathly Hallows also implies that it is specifically the act of creating Horcruxes that makes the soul so unstable:

“And the more I’ve read about them,” said Hermione, “the more horrible they seem, and the less I can believe that he actually made six. It warns in this book how unstable you make the rest of your soul by ripping it, and that’s just by making one Horcrux!”

If this is true, then Muggles should not be able to qualify for the inadvertent "Horcrux" situation. This is because even though they can rip apart their souls by murdering, they can't render their souls entirely unstable by making other Horcruxes (as that requires a spell).

Additionally, the piece of Voldemort's soul seems to have only latched on to Harry because Voldemort couldn't die (because of his other Horcruxes). But if someone who could die was in the same situation, they would just die. If the mere fact that they murdered would be enough to keep them alive then all murderers would be immortal (at least until whoever possesses their soul dies), something which we do not find in any of the books.

Or perhaps put differently: When a regular murderer (with no prior Horcruxes) gets killed he simply dies. Even though his soul has been ripped apart because of the murder, there is nothing tethering his soul to this world, so his soul would simply do what anyone's soul does when they die. However, if someone had a prior Horcrux then when he gets killed he won't die, because the Horcrux is tethering his soul to this world. In such a case there could be the potential for a piece of his soul to break off and latch on to another living being. However, it still seems that this would not happen to someone who only had one Horcrux. It seems from Dumbledore's statement that Voldemort's soul was especially unstable because of the multiple Horcruxes.

One could also argue that there is no fundamental difference between a soul split simply from murdering, and a split soul that was turned into Horcruxes. The reason then that Voldemort's soul was so unstable would be the multiple murders rather than the Horcruxes themselves.

In any case, this type of situation shouldn't be able to happen to a Muggle because a Muggle wouldn't have a prior Horcrux to tether his soul to this world. Even if his soul was rendered unstable it would not latch on to another being.

No they can't.

As Slughorn told Tom Riddle,

‘By an act of evil – the supreme act of evil. By committing murder.
Killing rips the soul apart. The wizard intent upon creating a Horcrux
would use the damage to his advantage: he would encase the torn
portion –'

‘Encase? But how –?'

‘There is a spell, do not ask me, I don’t know!' said Slughorn,
shaking his head like an old elephant bothered by mosquitoes. ‘Do I
look as though I have tried it – do I look like a killer?'

A muggle would not be able to use a spell so that rules muggles out. As for can it happen accidentally, it has been stated many times in the Harry Potter books themselves that what happened with Voldemort was a very exceptional case.

• he had multiple horcruxces (something that had never happened)


• his multiple horcruces had weakened the main part of his soul (so made it unstable)


• he may have been trying to create another horcrux at that very point


• Harry's mother provided him with a protective shield against Voldemort


• Voldemort was acting on a self fulfilling prophecy (he was marking the boy as his equal)

Considering all this, it doesn't look like accidental soul-splitting-off-and-attaching-it-self-to-the-only-other-living-being-in-the-room would even happen with another wizard

So I would say muggles are definitely out of the question

No.

We know that it is for sure that Horcruxes can be made from non-magic murder, i.e the diary.

The diary dates back to when Voldemort was still Tom Marvolo Riddle. After opening the Chamber of Secrets, he used the Basilisk to kill Myrtle Warren, providing the murder necessary to craft a Horcrux.
_{- Everyone you didn’t realise was connected to Voldemort’s Horcruxes - Pottermore}

I consider this non-magic because of the lack of a wand, some people might argue that he used parslemouth and somehow link that to magic but i disagree.

It is referred to as a Magic-Creation.

"Of the Horcrux, wickedest of magical inventions, we shall not speak nor give direction —"
_{- Horcrux - Harry Potter Wiki}

Now this is a quote from the book but i dont have a copy on me at the moment.
And this is another one.

"While the magical container is still intact, the bit of soul inside it can flit in and out of someone if they get too close to the object. I don’t mean holding it for too long... I mean close emotionally. Ginny poured her heart out into that diary, she made herself incredibly vulnerable. You’re in trouble if you get too fond of or dependent on the Horcrux."
_{- Horcrux - Harry Potter Wiki}

This quote leads me to believe there is some magic involved after the murder. And now that i've read @Alex's answer this quote seems to support that.

"By an act of evil — the supreme act of evil. By commiting murder. Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: He would encase the torn portion —"
"Encase? But how -?"
"There is a spell, do not ask me, I don't know!" said Slughorn,
_{- Harry Potter and the Half Blood Prince, Chapter 23 (Horcruxes)}

So, from this we can draw the conclusion that to transfer the piece of soul to a container some magic is required, and at this point it's redundant but Voldemort did make an inadvertent Horcrux but this was only possible because his soul was already so unstable.

When Voldemort attempted to murder the infant, a piece of his maimed soul attached itself to Harry Potter. As a result, some of Voldemort’s thoughts and abilities were shared with him, until Harry was struck with the Avada Kedavra Curse.
_{- Everyone you didn’t realise was connected to Voldemort’s Horcruxes - Pottermore}

From PotterCast’s JK Rowling Interview (quoted below) we know that creating a Horcrux needs to be done intentionally and requires (Dark) magic. The only reason that part of Voldemort's soul was able to leave his body and attach to Harry was because he had done the process so many times, each destabilizing his soul further, and then was hit with the backfiring of his own Killing Curse:

Here is the thing: for convenience, I had Dumbledore say to Harry, "You were the Horcrux he never meant to make," but I think, by definition, a Horcrux has to be made intentionally. So because Voldemort never went through the grotesque process that I imagine creates a Horcrux with Harry, it was just that he had destabilized his soul so much that it split when he was hit by the backfiring curse. And so this part of it flies off, and attaches to the only living thing in the room. A part of it flees in the very-close-to-death limbo state that Voldemort then goes on and exists in. I suppose it's very close to being a Horcrux, but Harry did not become an evil object. He didn't have curses upon him that the other Horcruxes had. He himself was not contaminated by carrying this bit of parasitic soul.

[...]

I do think that the strict definition of "Horcrux," once I write the [Harry Potter Encyclopedia], will have to be given, and that the definition will be that a receptacle is prepared by Dark Magic to become the receptacle of a fragmented piece of soul, and that that piece of soul was deliberately detached from the master soul to act as a future safeguard, or anchor, to life, and a safeguard against death.

Because creating a Horcrux needs to be done intentionally and requires magic, a muggle would not be able to do it accidentally. They would not be able to destabilize their soul enough without creating a Horcrux either, nor would they be likely to be hit with the rebound of a Killing Curse.

No - their soul won’t be unstable enough.

Soul pieces, even of people who’ve committed several murders, don’t just spontaneously leave their bodies. Wizards other than the Dark Lord who’ve murdered haven’t had their souls detach from their bodies and attach themselves to other people. Several of the Death Eaters have murdered, and their souls hadn’t attached to anything else - many of the Death Eaters were clearly seen being killed and no spirit form of them escaped, which couldn’t happen if there were any Horcrux-like pieces of soul that existed outside of their bodies.

“Well, you split your soul, you see,’ said Slughorn, ‘and hide part of it in an object outside the body. Then, even if one’s body is attacked or destroyed, one cannot die, for part of the soul remains earthbound and undamaged. But, of course, existence in such a form …”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)}}

The only reason the Dark Lord’s soul could have a piece detach unintentionally like that is because he’d already made his soul very unstable by making several Horcruxes.

“And the more I’ve read about them,’ said Hermione, ‘the more horrible they seem, and the less I can believe that he actually made six. It warns in this book how unstable you make the rest of your soul by ripping it, and that’s just by making one Horcrux!”
_{^{- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)}}

Dumbledore explained that’s why the Dark Lord’s soul split apart and attached itself to Harry - his soul was made extraordinarily unstable from creating several Horcruxes.

“He had rendered his soul so unstable that it broke apart when he committed those acts of unspeakable evil, the murder of your parents, the attempted killing of a child. But what escaped from that room was even less than he knew. He left more than his body behind. He left part of himself latched to you, the would-be victim who had survived.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 35 (Kings Cross)}}

That situation wouldn’t be possible for a Muggle who’s committed murder. Making a Horcrux requires a spell to actually encase the piece of soul in an object after committing a murder. Since removing and encasing the soul piece outside of the body requires magic, the Muggles wouldn’t be able to make even one Horcrux, so couldn’t destabilize their soul by making several of them in the way the Dark Lord did.

“Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: he would encase the torn portion –’

‘Encase? But how –?’

‘There is a spell, do not ask me, I don’t know!’ said Slughorn, shaking his head like an old elephant bothered by mosquitoes.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)}}

Though it’s likely that Muggles’ souls will be torn by committing murder, they won’t be able to destabilize their souls by taking the pieces and encasing them in external objects, and therefore it’d be no different than wizards who’ve murdered but haven’t created any Horcruxes - whose souls don’t detach from their bodies.