The bounce is the result of a stretch/contract reflex in your hamstrings/glutes. To get the "bounce", you need to:
1. Go below parallel. Hip joint below the knee as seen from the side.
2. Maintain an enormous amount of tension in your glutes/hamstrings, which requires you to use the proper mental cues, keep a tight arch in the back and proper knee position. To get a feel for this tension, work on unweighted box squats: in particular, take your normal squat stance and sit back down onto a box that is just below parallel. Do not let your knees move forward AT ALL. Then, stand back up from the box, again without letting your knees come forward AT ALL. Your shin should stay perfectly vertical the entire time. If you do this, you'll find yourself using your glutes and hamstrings a ton. This is not how you should actually squat, but the exaggerated motion and focus on the posterior chain should help teach you the proper "feel" of engaging your hip extensors (glutes, hamstrings, adductors).
3. Timing. If you pause too long at the bottom or try to get back up too quickly, you'll lose a lot of the bounce. One way to try to get the timing down is to do "one and a quarter squats" unweighted or with light weight: drop into a full squat, come back up a quarter of the way, quickly drop back down into a full squat again and then recover fully. The "quarter" squat in the middle lets you practice with the "bounce" feeling and is sometimes used by olympic weightlifters when they miss-time the bounce on the first recovery (such as in
this video).