IN MY OPIONION, he loves box squats because a). He helped popularize the exercise, and b). they carryover very well to the canvas suited squat that he and his lifters use in competition. A canvas suit "catches" the lifter on the decent, much like a box, the ipf poly suits "rebound" and stretch, hence the unpopularity of box squatting among those lifters.
I asked him if he were training an athlete who was a basketball player, and their main goal was to increase vertical jumping, what squatting means would he use. He said he would use a high box squat, with the box set at a little above a half squat, narrow stance, similar to the jumping position. This led me to ask him if he felt the stretch reflex was being detrained in a way from doing so. He responded that he had done extensive testing, and the stretch reflex could be maintained for several seconds in experienced lifters. The problem with this is the stretch reflex starts to dissapate immediately when the lifter hits the box, the longer the pause the greater the dissipation.
I like box squats COMBINED with free squats in a lot of cases, the box squat aids tremedously in different circumstances like teaching the squat, coming out of a dead start like a sprinter coming out of the blocks, and allowing a greater load when put in a higher position in the range of motion with a concrete indicator of depth. In squatting for jumping improvement, IN MY OPINION, the amortization phase, the point in the movement where the bar goes from downward motion to upward, is the single most important part of the lift. I have seen and strongly believe that the athletes who can turn the lift around the fastest will always have the best carryover to jumping. It is more about how quickly you can get to top speed at that particular point in the lift, than the overall speed attained during the movement.
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