There are many advantages to box squatting. One of the most important is recuperation. You can train more often on a box than you can doing regular squats. The NBA's Utah Jazz do box squats for the same reason - recuperation.
The second reason is equally important. It is generally accepted that you should keep your shins perpendicular to the floor when squatting. With box squatting, you can go past this point (that is, an imaginary line drawn from your ankle to your knee will point toward your body), which places all the stress on the major squatting muscles- hips, glutes, lower back, and hamstrings. This is a tremendous advantage.
Thirdly, you don't have to ask anyone if you were parallel. Once you establish a below parallel height, all of your squats will be just that -below parallel. I have seen it over and over. As the weights get heavier, the squats get higher. This can't happen with box squats. If our clients hips are weak, we use a below parallel box with a wide stance. If they need low back power, we use a close stance, below parallel. If their quads are weak, we work them on a parallel box. If they have a sticking point about 2 inches above parallel, as is common, then work on a box that is 2 inches above parallel. Our advanced squatters use all below parallel boxes. This builds so much power out of the hole that there will be no sticking points.
Now, how do you do a box squat? They are performed just like regular squats. First, push the glutes rearward as far as possible. With a tight back arch to descend to the box. Push your knees apart to maximally activate the hips. When sitting on the box, the shins should be straight up and down or even past perpendicular. This places all the work on the hamstrings, glutes, hips, and low back. These are the precise muscle groups that do a very large percent of the squat. After sitting completely on the box, some glute and hip muscles are relaxed somewhat. Then forcefully flex the abs, hips, and glutes and jump off the box. To ascend correctly, push the traps into the bar first. This will flex the back muscles, then the hips and glutes, and finally the legs. If you push with the legs first, you will be in a good morning position because the glutes will raise first, causing you to bend over. Always push the feet out to the sides, not directly down.
What about the development of power? Power is defined as work done divided by the time used to do the work. When you do a regular squat, you must do three things. The first is the eccentric phase, where muscles lengthen. When the eccentric phase stops, a static phase begins, where the muscles are not lengthening or shortening, but muscle energy is decreasing. Then to raise concentrically, you must start a load while the muscles are held statically, even to a brief extent. After all, power can be produced only so long. In a regular squat, you must produce power during all three phases, but a box squat breaks up the eccentric and concentric phases because some of the muscles are relaxing while others are held statically by movement in the hip joints. Here is where force can be redirected very strongly. Because a heavy squat uses a large amount of energy, it makes sense to break the work into separate parts. While box squatting is not plyometrics, it builds tremendous reversal strength.
Box squatting is not common, mostly because no one knows how do them. After reading this or visiting our facility you should be fully aware of the benefits and on your way to becoming more explosive.