ILLEGAL aliens are required to register with selected service. That is true but..
unless the rules changed since the 1950s when I served YOU HAVE TO BE A LEGAL RESIDENT TO SERVE IN THE MILITARY.
I don't think that the rules have changed. I recall a recent news item a few years ago where some Mexicans asked our Embassy in Mexico City if they could join our (U.S.) military. The answer was thanks but no.
I am not sure I remember all the wherefores and howdy doos, but my buddy’s German wife did not get citizen ship till many years after she had joined the military and did not meet him until years after she joined. Knew another dude who was Guatemalan or something he became a citizen while in.