There’s really no benefit to graduating early. All it’s going to do is put him at a one-year disadvantage at these hyper-selective schools he’s trying to get into. He’s competing against every other applicant with perfect grades AND went all 4 years. Even if he does manage to get in, the school is going to put restrictions on him, because he’s a minor. That’s not going to help if he already has issues with depression.
To me, it’s sounding like he’s looking for a panacea. If he doesn’t like going to the school he’s in, college is not the answer. Graduating early either isolates you from adult peers (not healthy for depression) in college, or relegates you to working at Subway for a year.
If he wants to get into a prestigious college like he’s telling you, then he’s far better off graduating on-time, so his application is competitive with all the other applicants. If this is indeed a panacea (I suspect it is) his real motive is going to be getting out of the boarding school, and he’ll start setting his college prospects lower and lower in order to escape(symptom of depression).
As a parent, this is all still your decision until he’s 18. Ask him questions and find out the motive. Starting college based on depressive behavior is not good. College stress will only make depression worse, even in the best of circumstances.