So the reason I do that is two fold. Since I am working on my squat max, I want to work the acceleration aspect of the jump through the entire range of motion of my squat. So kind of like a squat jump, but I land on a box. And since I was using it as a warm up for my squat work that day, I again was trying to focus on the range of motion I would be using later. Second, I once saw Shane Hammon (sp?) do them that way, and up until then I had always done them with just a quick counter-move. I thought "if this guy, who is an elite weightlifter, doesn them with a full squat, maybe I should try that."
I don't exclusively do them that way. I do three variations of box jump: Full squat, counter move, and with a run-up.
Full squat gives me more carry-over into my squat work. The counter-move has helped me get primed for Jerk work (I squat-jerk so there is some similarity), and I use ther run-up when I just want to go for max height, when doing "feeder workouts" where I am not doing anything for max weight, just going for CNS activation and speed.
I am probably somewhere in between fast and slow twitch dominant. Nature probably wanted me to be slow, but I have only trained to be fast, so its a bit hard to say if that is a factor. I am one of those people who has to jump off two feet, I don't get any height if I leap off one foot. I was a pretty good sprinter, but horrible at long and high jump (go figure on that one!).
Hope that clears up my thought process somewhat, let me know if you have any other questions.