What is the purpose of boxjumps in CrossFit?
The boxjump is the most common plyometric exercise in CrossFit. Even though the jump uses the calves, quats and the core, it is not meant for muscle build in the first place. Plyometric movements develope our explosive power - very beneficial for everything that requires the generation of power: Olympic Lifts, kettlebell swings, kipping ring-muscle ups ...
Should I rebounce?
Rebouncing a boxjumps means that you jump down the box and use the kinetic energy from the landing to push yourself up again straight away.
The obvious advantage of the rebounce is the speed. You can do multiple reps really quickly in a row.
On the other hand, the rebounced boxjump is more tiring for the legs. It is also not suitable for people with sensitive knees or ankle or calve problems. If you are not doing CrossFit on a competitive level, there is actually no point in doing the rebounce - at least not on a regular basis and for huge amount of reps.
You might even be unable to do it. It is like the doubleunders, you need to learn the rhytm and timing first. This video might help:
What if I canot overcome my fears?
Yes, the boxjump is a mental thing. Why actually every healthy person should have the strength to perform at least a 20 inch-boxjump, some people just find themself unable to perform the closed feed take-off and landing - because their head holds them back.
It can be a good start to just step up the box, see how you feel on top and then jump down. Just to get a feeling for the height (and to see that it is actually not high at all). Even better is to stack up some plates (take the heavy ones that won´t slip) and maybe start with 10 inch or 12 and slowly build up from here.
In most cases, it is only the very first boxjump anyway that is the issue. Once you know that you are capable of doing it, it will be downhill from there.
The #firstboxjump for some CrossFitter is not any less important than the #firstmuscleup.
Keep telling yourself: It is only a wooden box. It has no power over you!
What is a suitable scaling option for the boxjump?
Nothing can replace a jump - except a jump. While box step ups are a commonly used scaling option, it is actually a completely different stimulus. It is more a strength exercise that tagets the obliques and the VMO. The explosive component is completely missing.
So again: Jumping on a reduced height - 10, 12, 15 inch, whatever - will get the work done better.