Illegal aliens are not allowed to join the military. There is an expedited process for citizenship for green card holders who join the military. That said, they are only given limited security clearances. Certain kinds of intelligence are not permitted to be given to foreigners including green card holders.
You apparently know the hoops and ropes on such situations. However I am of the opinion many of those I was thinking about would be willing to be just ordinary grunt foot fighting men or of similar rank in the mechanized forces. Such I would have welcomed in WWII as opposed the fake soldier from Idaho.