Boards and boxes serve one main purpose and that is to break the concentric/eccentric chain by rapidly halting the bar during the eccentric portion. This leading to a build-up in kinetic energy resulting in a greater strength potential. Another benefit of these tools (boards for the bench press and boxes for the squat) is that you can manipulate with great precision the range of motion of the exercise. For example, by using three boards stacked together (called a 3 boards press) you decrease the range of motion of the bench press by about 6. This allows you to overload the finishing portion of the movement.
Some peoples claim that simply performing partial lifts ( half-squat, half-bench press) will do the same thing. Not true! You see, during a partial lift you have to decelerate the bar voluntarily before lifting it, this is good to develop deceleration strength and even isometric strength, however it is of limited use when trying to increase acceleration and power. To maximally increase those we need to develop our capacity to use kinetic energy to our advantage. The boards or boxes rapidly halt the eccentric portion without you significantly decelerating the bar, this results in a tremendous build up of kinetic energy which can be used during the subsequent concentric action. Basically board presses and box squats are very similar in effect to high intensity plyometric training.
For maximum effectiveness, you cannot stop on the board/box for more than 2 seconds, time after which the gained kinetic energy will be lost. Ideally the contact time would be shorter than one second, but you must still produce a lot of kinetic energy by avoiding simply patting the board/box; you must try not to decelerate too much: act as if the board/box didn’t exist, lower the bar normally until the object suddenly stop the bar descent.
Obviously we don’t what to exaggerate andslam into the board/box either. This will not bring any additional benefits and could actually increase the risk of injuries.