If I was a parent of a collegiate athlete, I'd flip my shit if I found out the coach was asking random people on the Internet advice for dealing with my kid's injury. It's one thing to look for advice and second opinions from peers, but this just seems scary. (No offense to the guys who've replied. I'm sure you get my drift.)
With that said, I suggest you seriously consider the pros and cons of having injured or recently-injured athletes squat, and look at all possible alternatives. You're still being vague about the actual injury. Is it a herniated disc, torn low trap, a sprained rhomboid? The particulars will obviously affect the entire training program, not just one lower body exercise.
If back squats are painful and front squats are "only a little better", then neither of them should be considered. You need to bring the upper back and spinal stress to a bare minimum. Something like a hip belt squat should help, but unilateral training would be my preferred choice. For the kid with lower back problems, a neutral grip (like a trap bar or dumbbells) should also be doable, but I'd also rely on unilateral stuff.
Have you been having this kid do front squats since last year even though the pain is "only a little better" with them?