Get all you can while you're still "fresh". However, I heard a statistic the other day that the pay difference for someone with a degree from a State school is not much different (not much, if any, lower) than the same degree for someone that went to an expensive private college.
If I'm understanding this right, you are talking about getting a highschool diploma from a private boarding school. That should increase odds on getting into a prominant or prestigous college but I doubt it would be much, if any, real benefit over a regular HS diploma from the public school system if your education stopped there. Yes, it shows you have discipline and structure but it's still only a highschool education even if it's a glorified one.
What I'm getting at is, go to school (HS and college) and I probably wouldn't worry so much about where exactly you get the degree but just rather that you get one. Yes, a better school may help a little bit, but it might not help out $100K (price difference for education at state college versus private) difference. Just a little something to consider. None the less, get the education. Regardless of where you choose to get it from, just get it.
C-ya