Ummm...if you look at the National Guard and Army Reserve, you will find that many of their full-time employees do, in fact, belong to a union. There are two kinds of full-time support Soldiers, technicians (General Schedule and Wage-Grade) and AGRs (Active Guard and Reserve). AGRs were formed because: the unionized technicians didn’t think they should wear uniforms; didn’t think they should have to go to weekend drills or the two week Annual Training; didn’t think they should deploy with their units to combat zones; didn’t think they should belong to the National Guard or Army Reserve.
You are either a sworn in soldier — active, guard, or reserve — or you aren’t a soldier. There is no such thing as a soldier who is not sworn in.
Any soldier who has a civilian job, whether government or private, can belong to a union for THAT job. But not for his military unit of assignment/duty.