How do you feel about them? I know powerlifters seem to swear by them. I've just never been a fan. Most people I see do them and rest. I don't think you're supposed to rest any part of the body, especially the back/lumbar, correct? Or perhaps the legs should rest and this is the goal?
I see many people rolling the spine and leaning back. I don't get it. It seems that a deep, slow,controlled squat with a pause at the bottom will be just as good. Granted the lower chain, etc having to fire with no momentum is great. For basic and even advanced training I don't see the reasoning. I would put them in a "specialized" category.
convince me.
An inquiring mind is a good thing. I don't think I'm going to convince you of anything, nor do I wish to (I personally don't utilize box squats "very often."), but I can satisfy some of your inquiries for the sake of answering a good question.
On a more superficial level, the general perception is that box squats are meant for specific sticking point training, which it can be, though it'd only deal with the upper half of the ROM. Theoretically, you would use more weight then you are able to go HtC and take advantage of the shortened ROM to acclimate to more weight and to train specific weak points of the ROM.
On a deeper look at box squats, what you're primarily training are motor units. Before the muscle fibers contract, the neurotrasmitter acetylcholine will activate your motor units. Motor units innnervate an "X" amount of muscle fibers. A motor unit could innervate 10 muscle fibers are 1,000 muscle fibers. Once your motor units are activated, the muscle fibers that specific motor unit innervates will shorten (contract).
A few neural adaptions occur with a complete stop, into a full contraction movement (ie.. box squats).
#1. You can train your motor units or better put, cause an adaptation, where your motor units innervate a greater number of muscle fibers. Imagine a hand that controls 5 fingers to suddenly close and make a fist (ie. contract), to a hand with 15, 20, 100 fingers at its command to close and make a fist.
So know imagine your quad and along your quads are muscle fibers and spread out across these muscle fibers sits a motor unit, thousounds of motor units, each with their own authority to command the contraction of a given amount of muscle fibers.
Here's where the real strength comes in.
#2. You can train your motor units to perform "Coordinated Motor Unit Firing," which is exactly how it sounds. When your muscle contract, say in a squat, not all of your muscles contract at the exact same time. Motor units are activated in as needed basis (which is also a principle as to why we really warm up; its not just for blood flow, but to recruit motor units), so here's an analogy to normal bodybuilding muscle contraction. Imagine a giant city and its completley blacked out. Now imagine if lights started to come on across the city, relatively at the same time, but realistically, it 'd look more like one light here, two lights there, five, six lights there until all the lights are on. Metaphorically, that's how our muscle normally contract. One motor unit activates, then two, then five, then fifty..... all within seconds of course. This is why the weight feels lighter as we warm up, because we have activated many motor units at this point, but they don't all remain activated until required.
Now, with a highly adapted Coordinated Motor Unit Firing environment, imagine that same blackened city and, now upon time of contraction, all the lights turn on at the exact same time. When all the motor units activate at the exact same time, instead of in progressive spurts, you can the ability to drive with much, much greater power.
Here's the whole picture, complete stop and go training causes your motor units to innervate a greater amount of muscle fibers and along with that, causes your motor units to firing synchronously, which makes your contraction much more powerful.
In the long run, the increased strength benefits will augment the hypertrophic benefits.
Please feel free to ask me for clarifications if need be and I apologize for any spelling and gramamtical errors, its a big post and I didn't proof read.