Videos of your squats would help, but I'm not about to suggest you go out and squat. No point in injuring yourself to find out how you got injured
Your description of military press scares me as well. You should not be over arching your lower back to get your reps. If your back is giving out or you cant keep an arch (not just for this, but any exercise that requires it) then either stop or reduce the weight.
Depth doesn't matter, what does matter is how good your form is at that depth. If you can't do a deep squat with good form, then don't. Only go as far as you can with good form and work on getting better in a deeper position, but not with a working sets weight.
Q&A:
How heavy is your squat?
Where was the bar positioned? (on traps, lower than traps, back of your neck, etc)
What were you looking at? (floor, ceiling, hot chick doing lunges, etc)
When you squat, do you push with your heels or your toes?
Where does your chest point? (down, up, sideways, etc)
More to come after these are answered
Others will probably chime in as well.
Just to be clear up front, I am in no way going to attempt to diagnose or deal with the back pain. My aim is to dissect your squatting and help with form.