Whether you realize it or not, you are what we call, "quad-dominant". What does this mean? Basically, when you perform any variation of squat or deadlift, you tend to rely on your quads to do the majority of the work - not real strong. Ideally, when we squat or deadlift, the primary movers should be the glutes, hamstrings, and lumbar, which acts as a stabilizer - we call this the Posterior Chain, and it is the high-horsepower engine of human movement...or at least it should be.
Before we start to breakdown the box squat, it's important to understand one idea that is directly correlated to the box squat movement, and that is that, "Wide always transfers narrow, but narrow does not transfer wide". Simply put, if we were to train your squat and deadlift using only wide stances, your standard(and even super narrow)squat and deadlift would improve drastically. Where as if we flip that and train only narrow stance squats and deads, then move to a wide set-up, your lifts would suffer quite a bit. Make sense?
Now, back to the box squat. As you can probably guess, this squat variation requires a box. "What height", you ask? Well, this is kind of a tricky question. It depends on what range of motion you are targeting, but for the most part you should find a sturdy box that allows you to get down to parallel or even 1" below parallel. Once you have that established, it's time to get organized.
This next point is something that is often missed by lifters, and when neglected, it severely decreases the power production of the squat. What is this magical secret? You MUST get organized BEFORE loading your body with weight!!! So many lifters lazily set themselves up and simply stand up with the bar on there back. When moving heavy loads, there is no way that you can become "tight" and organized with that load resting on your frame. So, that being said, you have to un-rack the weight using a good, strong position from the go.
I always cue my lifters to act as though they are drawing in a deep breath through a very small straw. This violent intake of air will ensure that your lungs, and therefore, you chest and abdominal cavities are solid. As a side note, you will notice that whenever you are forced to move anything heavy, you tend to create pressure by holding your breath, right? This is a safety mechanism that we've been born with. It's the mainstream exercise "experts" that recommend exhaling while exerting effort. That's silly. Don't release all of your air unless you want to turn into an empty sack.
Next, the stance in the box squat should be one that is well outside of shoulder-width. In fact, you should go as wide as possible while maintaining a "knees out" or "knees over ankles" position. We all know by now that if your knees collapse inwards, you lose all power production. So let's NOT do that!!!
After establishing your stance, your focus now turns to(once again) getting tight. As you descend to the box, you must focus on tempo. If you rush the down phase, there is no active "winding-up" of the primary muscles. This results in the lifter being crushed by the load because they beat the weight to the bottom and are now attempting to go back up while the weight is still traveling down. Not a good scenario! So, to avoid that, the down phase needs to be slow and controlled so as to give the lifter time to "wind-up" the primary movers in order to create as much torque as possible in the bottom of the squat.
The key components in this "wind-up" are driving the knees out as much as possible, sitting the hips back as much as possible, and maintaining a lumbar curve. Do this correctly, and you should reach the top of the box(bottom of the box squat) in a perfect position to pause....and then regenerate force for the up phase.
As you can probably tell, your goal in the box squat is to get wide and sit back! But, why is this? We've already hit on the benefits of wide stances versus narrow, but why is it so important to get back on the box? Well, if you remember, you are quad-dominant. By sitting back, we are forcing the posterior chain to take on the majority of the work load, which is the way it should be!
When we begin to stand back up, we have to PULL ourselves into the mechanically-advantaged groove. This can only be accomplished by engaging the hamstrings. Once we hit that groove, then we aggressively accelerate upwards to full hip extension, all while relying on the posterior chain to get the job done.
As always, with a new movement, this is a skill that should be practiced with very light loads first! In fact, it would be hugely beneficial to have a knowledgeable coach supervise the session to ensure proper form and technique. Other than that, the box squat is a movement that you should definitely add to your routine!