Nope, they would need to sign new contracts, and take new oaths. That can not be legally forced.
Well, that would depend how they do it, wouldn't it?
The United States fought under the flag of the United Nations in Korea; did those soldiers have to sign new contracts? Do the soldiers who go on the officer exchanges between the US and its allies have to sign new contracts?
It seems like there's plenty of precedent to say that American troops can fight under UN control and can be made to obey orders from superiors who aren't part of the US armed forces. I'm not sure why ceding military control of the US armed forces to the UN necessarily has to mean anything more than that.
Edit: but here's a hypothetical. Say you're a soldier when something like this transpires. Congress and the President say it's legal; you disagree. Do you have the right to desert? What about on the eve of battle between UN forces (the US included) and some enemy?
And here's the bonus question: if your answer is "yes", how is this fundamentally different from a soldier deserting because he feels the Iraq War is illegal?