I started to reply to this message when you posted, but was unsure of your background...... I was confused as to why you would be jump squatting more then you are squatting. Then I saw the video demo that you were emulating and didn't post. But, having said all that, I would have said the following:
Nothing should hurt you. Period.
Start light, and work your way up. That will help you determine just how to land. Yes, the forefront of your foot should hit first, followed by the entire foot. The lowering should be like you are squatting, and should be a natural, and ingrained motion. I can only imagine the knee pain of squatting on your toes, if that's what you were eluding to asking if your heel should hit the ground.
The question of leaving the ground, or flying, it depends on what YOU need to work on, and why YOU are doing this exercise. Meaning, in your quest of increasing power (the purpose of jump squats?), you may need more focus on either speed, or strength. A jump squat that is heavy, is probably not going to help you in the speed department. Neither will speed be helped if the jump squat is heavy and slow.
That's the video you are working from. That is a world class lifter, doing something specific to help their lifts. It's hard to say how you fit into that equation. I often do jump squats and use them with my clients, we use much lighter weights, and use a full ROM. Nothing at all what Aimee is doing in the video. And nothing that hurts the knees. Small note here, I don't do these with the majority of my clients, because a perfect squat is needed before I would have them jump and have weight on their back
Hope this gets you started......