Why didn't Rufus Scrimgeour escape the Ministry of Magic when he knew the Death Eaters were staging a coup on the Ministry of Magic? Why didn't he summon the Order of the Phoenix for help when he was captured by Death Eaters?

Scrimgeour's stances are sometimes debatable, but he's not a wimp.

Scrimgeour used to be an Auror.

Prior to becoming Minister of Magic, Scrimgeour used to be an Auror, and apparently not just an average one, as he was Head of the Auror Office:

"Sir, I saw in the Daily Prophet that Fudge has been sacked..."

"Correct," said Dumbledore, now turning up a steep side street. "He has been replaced, as I am sure you also saw, by Rufus Scrimgeour, who used to be Head of the Auror office."

"Is he... do you think he's good?" asked Harry.

"An interesting question," said Dumbledore. "He is able, certainly. A more decisive and forceful personality than Cornelius."

"Yes, but I meant --"

"I know what you meant. Rufus is a man of action and, having fought Dark wizards for most of his working life, does not underestimate Lord Voldemort."

_{Harry Potter and the Half-Blood Prince, chapter 4, "Horace Slughorn"}

Being an Auror requires a spine. He's probably been catching Death Eaters for a while, he is not going to turn back and leave now that he has even more power to do so. Talking of which...

Scrimgeour was (pretending) to fight Death Eaters as Minister.

After the Ministry admitted that Voldemort was back, they tried to make it look like they had the problem in hand. Admittedly, that went through bogus arrests such as Stan's (see below), but in any case, they were trying to do something - unlike Fudge, who was actively saying that everything was fine and that all that thing about "Voldemort's rebirth" were the ramblings of a teenager.

"'Stanley Shunpike, conductor on the popular Wizarding conveyance the Knight Bus, has been arrested on suspicion of Death Eater activity. Mr. Shunpike, 21, was taken into custody late last night after a raid on his Clapham home...'"

"Stan Shunpike, a Death Eater?" said Harry, remembering the spotty youth he had first met three years before. "No way!"

[...]

"Yeah, that's him," said Harry. "I dunno what they're playing at, taking Stan seriously."

"They probably want to look as though they're doing something," said Hermione, frowning. "People are terrified--you know the Patil twins' parents want them to go home? And Eloise Midgen has already been withdrawn. Her father picked her up last night."

_{Harry Potter and the Half-Blood Prince, chapter 3, "Will and won't"}

Scrimgeour didn't apparently know that his Ministry had been infiltrated.

The "coup" was a planned operation, meant to last, unlike the hit-and-run operation to kill Dumbledore. It took the Death Eaters a good time to get the right man under their influence (Thicknesse under Imperius Charm)...

‘Well, Yaxley?’ Voldemort called down the table, the firelight glinting strangely in his red eyes. ‘Will the Ministry have fallen by next Saturday?’

Once again, all heads turned. Yaxley squared his shoulders.

‘My Lord, I have good news on that score. I have – with difficulty, and after great effort – succeeded in placing an Imperius Curse upon Pius Thicknesse.’

Many of those sitting around Yaxley looked impressed; his neighbour, Dolohov, a man with a long, twisted face, clapped him on the back.

_{Harry Potter and the Deathly Hallows, chapter 1, "The Dark Lord Ascending"}

And even then they had to wait before that man himself converted the rest, so that Scrimgeour had no way of escape.

‘It is a start,’ said Voldemort. ‘But Thicknesse is only one man. Scrimgeour must be surrounded by our people before I act. One failed attempt on the Minister’s life will set me back a long way.’

‘Yes – my Lord, that is true – but you know, as Head of the Department of Magical Law Enforcement, Thicknesse has regular contact not only with the Minister himself, but also with the Heads of all the other Ministry departments. It will, I think, be easy, now that we have such a high-ranking official under our control, to subjugate the others, and then they can all work together to bring Scrimgeour down.’

‘As long as our friend Thicknesse is not discovered before he has converted the rest,’ said Voldemort. ‘At any rate, it remains unlikely that the Ministry will be mine before next Saturday. If we cannot touch the boy at his destination, then it must be done while he travels.’

_{Harry Potter and the Deathly Hallows, chapter 1, "The Dark Lord Ascending"}

We don't know exactly if that "next Saturday" deadline is respected, but what we know is that on August 1, 1997,

"The Ministry has fallen. Scrimgeour is dead. They are coming."

_{Harry Potter and the Deathly Hallows, chapter 8, "The wedding"}

Read that as you want, but I personally get the feeling that Scrimgeour, and the 2-3 people that may not have been converted at that point, were overwhelmed very fast, with little to no time to Disapparate/fly away - even then, Death Eaters have the noble habit of firing on people's backs.

It's rumored that Scrimgeour was trying to protect Harry anyways.

When Lupin comes to Grimmauld Place, he offers more detail about the operation on the Ministry:

“Well, Kingsley saved us,” said Lupin. “Thanks to his warning most of the wedding guests were able to Disapparate before they arrived.”

“Were they Death Eaters or Ministry people?” interjected Hermione.

“A mixture; but to all intents and purposes they’re the same thing now,” said Lupin. “There were about a dozen of them, but they didn’t know you were there, Harry. Arthur heard a rumor that they tried to torture your whereabouts out of Scrimgeour before they killed him; if it’s true, he didn’t give you away.”

Harry looked at Ron and Hermione; their expressions reflected the mingled shock and gratitude he felt. He had never liked Scrimgeour much, but if what Lupin said was true, the man’s final act had been to try to protect Harry.

_{Harry Potter and the Deathly Hallows, chapter 11, "The bribe"}

Scrimgeour did try to save the appearances sometimes. He behaved like a spoiled brat, thinking he had the right to know everything that was going on, that Dumbledore could have told Harry, and he never knew. But still, the man stood his ground and went down protecting "the insolent and insubordinate" one. I say, respect earned!

In addition to the above, I first addressed the "not calling out the Order" by pointing out that the Order was a secret organization, but further reading shows that well, the Order indeed used to hide itself from the Ministry...

Lupin glanced at Harry, then said to Tonks, 'What were you saying about Scrimgeour?'

'Oh ... yeah ... well, we need to be a bit more careful, he's been asking Kingsley and me funny questions...'

_{Harry Potter and the Order of the Phoenix, chapter 7, "The Ministry of Magic"}

... though it's worth noticing that Scrimgeour wasn't yet Minister at that point, Fudge was. As a reminder, Fudge was completely paranoid of Dumbledore trying to overthrow him, so they obviously weren't going to announce themselves yet.

I say "yet" and "used to", because, times of war and all, it appears the Order eventually joined forces with Scrimgeour's Ministry:

“At the home of one of the Order,” said Snape. “The place, according to the source, has been given every protection that the Order and Ministry together could provide. I think that there is little chance of taking him once he is there, my Lord, unless, of course, the Ministry has fallen before next Saturday, which might give us the opportunity to discover and undo enough of the enchantments to break through the rest.”

_{Harry Potter and the Deathly Hallows, chapter 1, "The Dark Lord Ascending"}

Scrimgeour supposedly knowing about the Order, still does not mean he would have been prone to call more people to be slaughtered, nor may have had the time to do so.

In any case, "summoning" people isn't that much of a thing, at least not with Accio, if that's what you were thinking. Accio only works on inanimate objects (quote snatched from this answer):

Why couldn’t Newt use ‘Accio’ to retrieve all his beasts?

‘Accio’ only works on inanimate objects. While people or creatures may be indirectly moved by ‘Accio-ing’ objects that they are wearing or holding, this carries all kinds of risks because of the likelihood of injury to the person or beast attached to an object travelling at close to the speed of light.