Other mistake: At the end of basic training, Judy's platoon goes to the bulletin board to see what their M.O.S. will be. This is not how the U.S. Army operates, you must pass a qualification test for the M.O.S. that you enlist for before you ever enlist. You know before you ever get to basic training what M.O.S. you will be training for, not on the last day of basic training.
Other mistake: When Judy leaves basic training she goes straight to the Thornbirds as an 11-Bravo (infantryman) with an airborne identifier. She would have had to attend 5 weeks of AIT and Jump school (and no she couldn't have had her airborne training with the unit, Jump school is only at Ft. Bragg, N.C. or Ft. Benning, GA.) before joining that unit. The same mistake is made when she goes to SHAFE, Belgium (Supreme Headquarters Allied Forces Europe), presumably as a 71-Lima (personnel actions specialist) again without going to AIT.