You see it a lot on social media. People load up a bunch of weight on a bar, squat down until their butt touches a box or bench, then they stand back up to finish the rep. This movement is very unique in that it looks like a back squat, but with a varied range of motion depending on the height of the box or pins on the rack. Sometimes you'll see a pause on the box, or a touch on the pins and go. But any way you slice it, this movement is a variation of the squat. The question then becomes, "When do I utilize something like a box squat?".
First we need to define the type of barbell lifts and weight room movements out there. You have 3 types to choose from: Primary, Supplementary, or Accessory.
Primary:
These are the gold standard of barbell exercises. They range from the squat, bench press, standing press, deadlift, clean, and snatch. These movements are done normally for general strengthening exercises, or for most lifting competitions. These lifts are primary because they use a lot of weight, a lot of muscles, and train a fairly long range of motion. As far as "compound lifts" go, these are the most foundational to strength and conditioning.
Supplementary:
These exercises are variations of the primary lifts. They train a specific range of motion (either shorter or longer), or are a change in protocol on how the rep is completed. For instance, a supplemental deadlift movement could be a Rack Pull (reduced ROM), Stiff Legged Deadlift (Longer ROM), or a Halting Deadlift (Isometric hold at the knee). A supplemental movement for something like a power clean might look like a Hang Clean, a Block Clean, or a Halting clean. Either way you cut it, these exercises serve to increase performance on your main lifts. The marker of a supplemental lift is that it looks similar to a Primary Lift, but is set up or completed differently. These are used as tools for training around injuries, or training past a sticking point on certain lifts.
Accessory:
These movements are pretty much anything else in the gym. These exercises serve not to train a movement, but are used to train a muscle or a chain of muscles assisting in the main movements. These are your typical movements like Chin Ups, Triceps Extensions, Hamstring curls, etc. These serve to increase work capacity of a muscle involved in the primary barbell movements. When you put this into your program, you must do it sparingly. Too much accessory movements, and you run the risk of stalling or reducing your performance on the main lifts.
So where would this put the box squat? Well this would make it a Supplemental movement. There is a lot of eccentric stress that goes into the squat, you can reduce it by reducing the distance the bar has to travel. However, because you are reducing the range of motion, you must adjust the stress of the movement by increasing the weight on the bar. Because the volume of work has decreased (aka the bar moves less), you must supplement the work by adding more weight to the bar. There are circumstances when you would utilize a light weight box squat. For instance, a client with a condition or injury. As it was previously stated though, the supplemental exercises serve to increase performance of the primary lifts by addressing weakness or sticking points. If the client can't reach depth, a pause squat will help them find that position and strengthen it more effectively. If the client fails a rep on squat, during the ascent about 2 inches from bottom position, we've now found the sticking point on the lift. We can now use a box set 2 inches higher from the bottom position, to continue the work the squat without failing any more reps and still drive the weight up on the squat. Just training a pause squat or box squat for the sake of complexity or "switching it up" is not a rational approach to training with supplemental movements.
In essence, the box squat is a tool for training the squat. With the conjugate system we use, we box squat a lot to keep guys ready for practice and competition. We utilize the Box Squat for both Dynamic and Max Effort sessions. The goal of the session may differ in the types of movements used, but the training goal is drive progress on the Primary Barbell Movements.