Do you have The Barbell Prescription? It has two options for increasing range of motion for trainees that are too weak for a full depth body weight squat. I know you're not dealing with optimal equipment at your boxing gym, but do they have boxes? Enough plates to stack? Can't remember if the rack you squat in is adjustable enough for you to use the bungee method. You can LP range of motion if you get creative with the equipment available to you. Once you get him squatting to depth with body weight, loading it with dumbbells (which I assume your gym has) as a goblet squat is probably advisable before jumping straight to the bar.
BBRx also has a deadlift-centric program for those who can't squat. It calls for up to 3 sets of 5 on the deadlift, and that's for much older trainees. If the deadlift is going to be your son's primary strength builder, I'm sure at 16 he can recover from 3x5 for a good while. I don't know that deadlift weight will be a great predictor of when he can squat with the bar though. While it trains all the same muscles to a degree, it leaves out the range of motion he can't currently attain. There are ways to turn a deadlift into more of a squat pattern, but if he already can't squat I'm not sure if these would be a better option.