This may explain:
Certain non-citizens can enlist in the United States Armed Forces. To be eligible to enlist, a non-citizen must:
(1) Entered the United States on a permanent residence visa or has an Alien Registration Receipt Card (INS Form 1-551/I-551 greencard or stamped I-94), and
(2) Established a bona fide residence, and
3) Established a home of record in the United States.
The visa and/or "greencard" must have sufficient time remaining on it (expiration date) to be valid during the entire term on enlistment. While non-citizens may enlist in the U.S. Military, they are not allowed to reenlist (stay in beyond their first term of service), unless they first become U.S. Citizens. However, there are accelerated citizenship procedures for non-citizens on active duty.
Anyone who has been a resident of a country considered hostile to the US has to have a special waiver in order to serve in the US military. And, note the 'permanent residence visa' req'mt. That's a different visa from those that are granted for political asylum.
I understood thirty years ago that a foreign national could enlist at a US Embassy to start the process. That was how a lot of Filipinos joined the service.