It's important to understand how to perform a box jump. In my observation, most people are doing it wrong. In the today's world of Instagram and PR's (personal records) on video, box jumps have been taken to a whole new stupid level. The point of box jumps is to develop explosive full body power.
In many of these jumps where people are showing their vertical glory, though impressive, are exhibiting more so how high they can get their knees than actually lift their center of mass.
To do the basic box jump, you start in an athletic position. You initiate the movement by driving your hands down and back (keeping elbows as close to 90 degrees as possible or you'll break your fingertips either bringing the arms down or up) while loading your hips. The cue's I use are as follows:
1) Jump high, land soft: This cues power intent and controlled deceleration on to the box.
2) Get to the top of your jump as quickly as possible: This cues power intent. Many times in sports it's not how high you jump, it's how quickly you get to the top of your jump.
3) Land in the same position as you started: This ensures you are focusing on moving your center of mass vertically, not just how high you can get your knees up and land in a deep squat position.
4) Land as if you were dropping through a hole in the center of the box: This cue is to focus on jumping and landing with more vertical intent. Too much horizontal movement in a box jump can make for a shaky landing. The feet will hit the box and the forward momentum can tip the box over with you on top.
5) Step down off of the box: Most people do not need to do repeat box jumps (dropping back off the box and immediately returning to the box). The plyometric movement can be left for athletes with the dead set focus on improving the speed of their neuromuscular system. Keep it safe and step down, reset, and do the next jump.
Lastly, make sure you are using a box made for box jumps. Don't stack plates on boxes. Plates are for lifting, not jumping on (as you can see in this video) . Aerobic steps are okay if you are a beginner and only using 3-4 risers. But trying to be impressive with more than that can end up in a complete disaster like this guy trying to be awesome in this video.