MacGyver ran from 1985 to 1992. Gulf War I (Western Allies force Iraq out of Kuwait) Lasted from August 1990 to February 1991, so, if it involved the US being in an armed conflict, it certainly wasn't Iraq for MacGyver's backstory, if anything it would be a S6 or S7 episode
I know that it can't be Desert Storm with MacGyver. But I think it was mentioned that he was on some kind of a mission in Iraq during the Iran/Iraq-war. I know that there's at least one episode where he was in Afghanistan, but as I said, I remember something about Iraq in his background as well...
It wasn't in an armed conflict the US were overtly in - I never got the impression that MacGyver was a soldier. But it could have been a reconaissance mission or a mission to retrieve something.