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In Plato's Republic, a simple shepherd finds a magical ring, places it on his finger, turns invisible and then immediately sets about a campaign of murder, rape and treason. This seems like behaviour that would be out of character for anyone who isn't a Lannister.

Does the Ring have a specific corrupting influence (like Tolkien's 'One Ring' corrupts those who wear it) or is it simply Plato's opinion that anyone who can become invisible is basically mere hours away from being a murderous rapist?

It does not. And Plato doesn't actually commit himself to a position on the subject. Instead he puts the story of the ring of Gyges into the mouth of his brother Glaucon who argues that being placed in a position where one is no longer answerable for ones deeds, very few if any people would refrain from abusing such a position of immunity. They might not murder, but would they steal or spy? Would they behave in every way the same as they would if they were still to some degree answerable for their actions? Glaucon doubts it. But Plato also has Socrates say that the man who refused to use such power would be a happier and freer man while the man who used it would be a slave to his appetites.

But the philosophical issue being discussed is invalidated if the ring also has some kind of mind control power. The only corrupting power of the Ring of Gyges is the power it gives the user. I might add that it's probably a metaphor for being some kind of absolute monarch.

The past is a foreign country; they do things differently there.

^{L. P. Hartley, opening line of The Go-Between}

If you want to understand why Gyges behaves as he does, it's important to recognize the historical context of The Republic. Gyges's actions are unethical under a broad spectrum of "modern" ethical theories, such as these examples:

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  Utilitarianism ("The right thing creates the greatest amount of total happiness.") or consequentialism generally ("Morality is determined by the consequences of an action.")


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  Kantian ethics (very roughly "Don't use people as means to ends.") or deontology generally ("Morality is determined by our adherence to duties.")


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  Contractualism ("Society is an agreement not to harm one another.")


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  Virtue ethics ("The right thing exemplifies virtues such as honesty and integrity.")

With the exception of virtue ethics, none of these ethical theories were fully elucidated in Plato's time. There was, however, a rather different theory of ethics which people believed in:

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  Might makes right (exactly what it says on the tin)

Plato was arguing against might makes right, particularly in book one, where Thrasymachus quite explicitly makes this argument, but it is a continuous undercurrent throughout the work. This leads us to a rather startling conclusion:

If Gyges had not been depicted as a cartoonishly evil villain, some readers would have interpreted him as a hero, under the theory of "might makes right."

Why is that a problem? Many stories involve ambiguous or morally gray antiheroes. They're all the rage today, and have been in and out of popularity for thousands of years. But Plato was not trying to sell books. Plato was trying to make a philosophical point. And that point is not served if his readers get distracted and start a literary argument over one of his characters. Portraying Gyges as an unredeemable villain is necessary to make "might makes right" look bad.

It seems it's as simple as Plato believing that nobody who has power can avoid being corrupted if nobody is able to see them abusing it. So the ring would be as much a force for corruption as any other means of gaining power without any oversight.

For example: "If Plato's allegory of the ring is right, then we had better watch out. Anyone who gains power without accountability is liable to use it unjustly."
reference This is typical of the usual analysis of this allegory.

"The Ring of Gyges" is narrated by Glaucon, and immediately upon concluding the narrative he provides his commentary. At that point in the dialogue Socrates and Glaucon were discussing whether justice is something inherent, or whether it is merely a societal construct that people agree to in order to protect themselves from the injustices that others would inflict upon them. Glaucon argues that man would rather be unjust, and justice is a concession to man's own frailty:

They say that to do injustice is, by nature, good; to suffer injustice, evil; but that the evil is greater than the good. And so when men have both done and suffered injustice and have had experience of both, not being able to avoid the one and obtain the other, they think that they had better agree among themselves to have neither; hence there arise laws and mutual covenants; and that which is ordained by law is termed by them lawful and just. This they affirm to be the origin and nature of justice; — it is a mean or compromise, between the best of all, which is to do injustice and not be punished, and the worst of all, which is to suffer injustice without the power of retaliation; and justice, being at a middle point between the two, is tolerated not as a good, but as the lesser evil, and honoured by reason of the inability of men to do injustice. For no man who is worthy to be called a man would ever submit to such an agreement if he were able to resist; he would be mad if he did. Such is the received account, Socrates, of the nature and origin of justice.

^{(The Republic Book II translated by Benjamin Jowett; my emphasis)}

To this end Glaucon tells the story of Gyges as an illustration of the fact that anyone who had nothing to fear from other people would discard justice, having no need for it. Since the only point of justice is to protect one's own interests, if one is so powerful that no one else can harm him then the person would revert to the ideal mentioned above: "the best of all, which is to do injustice and not be punished". This is how Glaucon explains it:

Suppose now that there were two such magic rings, and the just put on one of them and the unjust the other; no man can be imagined to be of such an iron nature that he would stand fast in justice. No man would keep his hands off what was not his own when he could safely take what he liked out of the market, or go into houses and lie with any one at his pleasure, or kill or release from prison whom he would, and in all respects be like a God among men. Then the actions of the just would be as the actions of the unjust; they would both come at last to the same point. And this we may truly affirm to be a great proof that a man is just, not willingly or because he thinks that justice is any good to him individually, but of necessity, for wherever any one thinks that he can safely be unjust, there he is unjust. For all men believe in their hearts that injustice is far more profitable to the individual than justice, and he who argues as I have been supposing, will say that they are right. If you could imagine any one obtaining this power of becoming invisible, and never doing any wrong or touching what was another's, he would be thought by the lookers-on to be a most wretched idiot, although they would praise him to one another's faces, and keep up appearances with one another from a fear that they too might suffer injustice. Enough of this.

^{(The Republic Book II translated by Benjamin Jowett; my emphasis)}

Thus, the answer to your question is that the ring does have a corrupting influence on its wearers. But it is not a "magical" corrupting influence. The ring (i.e. the power to do what you want and not get caught) simply removes the utility of justice, which leaves man in his natural state where injustice is the best policy.