These military backs are really hard to remove. If you do manage to struggle one off, putting it back on ain't that easy, either. I'm sure that there are proper tools out there that will do the job perfectly but, if you're brave, here's how I do it.
First, using a small screwdriver or a plastic cocktail stick, put a few drops of your favorite penetrating oil around the join and leave it for a few hours.
This is the business end of my Dad's old mortising chisel. The blade section is about 7/16" square and the tool is about 18" long. The end will need a quite shallow bevel, less than that of a properly sharpened chisel.
Place the watch in a vise, which must have soft alloy protectors on the jaws or, better yet, medium hard plastic ones. Tighten the vise just enough to prevent the lugs opposite the chisel from riding up the jaw under pressure.
In this view you can see that the back does have a slight chamfer that the chisel blade can contact. Sometimes, the gasket is not fully compressed and there's a small gap to allow the blade to insert itself further.
Ignore the photo above, I don't push on the lug any more. Instead, I rest the chisel on the vice jaw, making quite sure that it is not going to bear on the strap bar. Push the chisel firmly forward into the chamfer on the underside of the snap-back and lever the chisel handle downward. The back can fly off with a loud ping, so arrange that it can not land good side down (like toast) on your hard uneven concrete floor.
If you need to remove the movement, push the little button while pulling on the crown
A woodworker's clamp can be used to refit the back to the watch.
Put a very thin coating of grease around the spigot on the body where the back fits. The PVC pipe connector fits over the bezel but does not press on the crystal. The scrap plastic plug presses on the inner part of the back - the theory being that the back might flex enough to open out it's edges a little.
Tighten the clamp slowly, all the while making sure that parts stay as parallel as possible. I find that pressure builds up and up - and then the back just slips on. Make sure it is even all the way around. Move the plug over and re-clamp if one side of the back is not all the way on.
Y'all be careful now and don't force anything!