I know this is extremely unlikely, but, in your opinion, what would the response from the U.S. and world be to a U.S. army turning traitor and taking over a nation such as Iraq or Afghanistan?
Rogue armies have been known to take over governments by coups d'etat; Libya and Argentina being cases in point. Other than a rogue general taking over Afghanistan or Iraq, you might further consider the possibility of it happening in this country. It is for this precise reason that our Constitution grants to the President the power of Commander in Chief. Any general who attempted to act on his own authority would be immediately relieved of command, and with no funds to pay his army or supplies to arm/feed them with, such a rebellion would end before it got off the ground. No commander in our armed services has the resources to equip his own army; and even those who have dared question the Commander in Chief have been out on their respective ears. Douglas Macarthur is a case in point: There is ample speculation that Macarthur hoped to be President one day, and began by openly criticizing President Truman, whom he had deliberately flaunted on more than one occasion. He soon found out what happens to those who act so irresponsibly. So to answer your question, if such a revolt occurred this morning, it would be over before we went to bed that night.
I'm not so sure that the United States wouldn't try to cover up such a scenario as you suggest. The rogue army's supply lines would be cut off--eventually--but I doubt that the American public would ever hear the real truth about such a caper. The government and military would put their own spin on such an operation in order to save face, and Americans would probably be quick to discount conflicting reports as propaganda.
I agree with Brett D that we would cut that army off. There would be very little support in the US for such an army and the government would easily be able to stop funding it.
The international community would, of course, condemn such an action. However, I do not think they would need to take any action whatsoever because I am quite certain the US would would cut off all logistical support to such an army.
Thankfully, I do not think that there is any way that this would happen. I cannot think what such a unit would think that they would gain from taking such an action.
It's difficult to assess exactly what you mean to ask with this question, but as you said, the chances of a US Army division or large unit of troops turning renegade and taking over a foreign country is unprecedented and probably impossible.
In the case of the Civil War, where whole states seceded and American soldiers defected to their new country en masse, the US government responded with force. I think if it happened in a foreign country, then the government would either send in new troops to try and assume command, and/or cut off supplies to the rebellious troops, meaning they would not be able to stay effective for long.