The primary reason for doing box squats is to strengthen the sticking point, weak point, or failure point a lifter hits while ascending from a deep heavy squat.
Squatting movements, whether done back or front, have three phases. These phases are the start, the sticking point, and the finish.
During the start of the squat the quadriceps are the prime mover. At the sticking point, the quads want to turn a lot of these prime mover duties over to the glutes and hamstrings. If the glutes and hamstrings aren't strong enough to take over at this point, the squat will fail.
Once past this sticking point, the quads will join in again with the glutes and hamstrings to finish the lift.
The spinal erector muscles during a squat are primarily stabilizers, not prime movers. The erectors act as prime movers during pulling movements.
In December of 1971, I personally discussed box squatting with Mr. Squat himself, George Frenn.
George was the architect of box squat routines. In the late 1960's, he became the first man to officially lift a double 800 (800 pound squat and deadlift together). This was done in the 242 pound class. In fact, George's total was higher than the best super-heavy at that time.
The 840 squat that George did weighed out heavy at 853. This was combined with a 520 bench and an 810 dead. George attributed this 853 squat to a low box/high box squat routine. He did this routine twice a week. Low box squats were done once a week and high box squats were done once a week on another day, usually on Tuesdays and Saturdays.
More, from the era, here:
The height of the box or bench was set 2 to 3 inches above or below the sticking point. Georges instructions to me and others who sought his advice was to sit on the box and relax the tension on the legs.
As an example, once while training with George, he asked me and another partner to spot him on a heavy high box squat. He had 805 on the bar. Having never spotted a lifter with such a heavy poundage, I was somewhat concerned. However, my concerns would be squelched quickly.
He took this 805 off the racks like it was 135. He stepped back with the box about one foot behind him. In a controlled manner he squatted to it. When his glutes touched, he sat erect. The angle of his back shifted from about 60 degrees to almost 90 degrees in relation to the box.
At this point what I saw blew me away. He lifted both feet off of the floor. After a brief pause he put his feet back to the floor, rocked forward, and rose for the first of two very easy reps.
I personally tried this high box/low box routine. I succeeded in improving my squat by 30 pounds in about 30 days. However, in retrospect as an Olympic lifter, I feel that this was a mistake. I thought at the time that the best way to get my clean and jerk up was to increase my squat. Increasing my squat 50 pounds did not increase my clean and jerk.
On April 12th, 1975, George Frenn represented my club at the California State Olympic Weightlifting Championships. As a superheavy weighing in at just 243, he won the class on his last clean and jerk with a 363 pound effort. In competition George used efficiently the old-style single pull split on his snatches and cleans. I frequently trained with him at this time and can at this time and can attest to seeing him consistently squat between 700 and 800 pounds.
I believe the reason George was able to win this contest was NOT because of his squatting prowess. It was probably due to the fact that he did a lot of power cleans, deadlifts, and good mornings. I personally saw him do over 500 pounds in the good morning.
If an Olympic lifter fails the rise after racking a clean it is pretty obvious that the individual needs to do more squats. However, as an Olympic lifter, there is such a thing as an excess of squatting power. It's always nice to have a little extra in the bank, but there is a saturation point that can be reached in relation to the clean and jerk.
There is a price to be paid for doing box squats as I have previously described. If one sits on an immovable surface with a heavy weight across the shoulders, the weight pushes straight down sandwiching the discs and vertebrae between the barbell and the immovable surface. The spinal erector muscles and the abdominal muscles are the main muscle groups that support and protect the spinal column. However, this support is primarily vertical for back and forth protection, not for up and down protection.
Constant compressing, squeezing, and crunching one's vertebrae and discs will more than likely lead to permanent damage over a period of time.
Regular continued back care exercises like hyperextensions, reversed leg raises and reverse hypers, and stiff-legged deadlifts could stave off this degeneration and extend one's lifting career.
A MODIFIED APPROACH that is less stressful is the wall of post squat. The principle here is that rather than sitting on something immovable, you lean against something immovable. This can be done by leaning against a low wall or an immovable post, or inside a doorway against the door frame. Two spotters are a must when doing this with heavy weights.
The first thing one needs to do when doing post squats is mark the floor so your stance is in the same place each time in relation to the post. When one is ready, descend in the squat to parallel. Once at parallel, one's butt should touch the post. Upon touching the post, pause, lean back and rest against the post. To recover lean forward and push the butt against the post to get started. This puts a lot of stress on the quads and is not for everyone. Sets and reps are up to the individual. However, warm up thoroughly first.
Another alternative to the box squat is the hack squat or hack lift. Depths can be determined by just standing on different height blocks. This is a great movement for strengthening the transition phase of the pull, from the first to second pull in a clean.
Also, squatting inside a power rack, from a deadweight position off of pins is good.
One's leverages and genetics will dictate the kind of gains one will get from these exercises.
However, the main goal should be safety, which translates to longevity.
Enjoy Your Lifting!