I would suggest you reread Minotaur's 3rd point a few dozen times, it is priceless (ok, I exagerrate slightly on the number of rereads needed). It does seem that how the squat starts is critical. I'd expect that the initial movement triggers subsequent muscle recruitment.
The other tip that I've found useful came from Dan John. The squat involves the body moving between the legs, not folding up like an accordian. Obviously (or maybe not) this would not be as applicable in the performing the box squat, which seems like it might be a good idea given what you hve described.
The three points of squatting I learned at an Elite FTS seminar (with Dave Tate and Jim "1,000 pound squat" Wendler) are:
1 - Grip the bar as hard as possible, that will tell the rest of your body to stay tight.
2 - Arch your upper back, which will keep your lower back arched as well.
And 3 - The first thing to move when descending is your butt. Your butt and hips should move back and THEN down. Think of doing a 1/4 or 1/2 good morning, then squatting.
That last point, doing a mini-good morning, might especially be worth a try.[/quote]