Have you read Aztec by Gary Jennings?
In terms of the initial question, the Marines win easily. The shock value of crude gunpowder in the middle ages was signifant. The firepower, mech, aircraft, and artillery in a MEU would be godlike to the Romans.
In essence, they'd become a super Praetorian guard that would rarely have to fire weapons after a couple of quickie encounters. Aimed fire taking out all centurions, tribunes, etc. at a few hundred meters would be impressive as well and would shatter a legion. Snipers using night vision taking out rebellious consuls, etc. None of that would require much ammo. A couple volleys of artillery fire would shatter a legion.
Romantic legends aside, virtually any troops will break and run if they believe that certain death lies ahead. And while the Marines would know their supply of weapons and aviation fuel wasn't inexhaustable, the Romans wouldn't.
The problem for the Marines is that their technology is so advances that they couldn't possibly duplicate it. You'd need some Master Gunnery Sergeants or warrant officers with hobbies in more primitive firearms to set things up. And of course, any MEU sized unit is going to include a fair number of guys with useful civilian-type skills.
I have not. I will look into it. Just recommended Siege of Malta by Fuller. Best military history I have ever read. Should be a movie.