My son is from a private military school and he did not get in. Two kids from two separate public schools did get in.
Our Mids attended a public high school. It's not so much public or private; it's caliber of the school. Our public high school is the equal of the very good private schools in this area. Some parents wanted one of ours to transfer to their private school for sports. He would have killed us if he had had to drop levels in competition. Also, I think they were both more competitive for USNA from their public school.
Finally, an admissions staffer told me that it doesn't hurt future applicants when their school has a history of grads doing well at the Academy. It gives USNA some idea that the school is rigorous enough to prepare them for success. I can name dozens of schools in this state that have never produced service academy kids and likely never will. When your valedictorian gets an 810 on the SAT, something's missing. Yep. I interviewed that kid. Really liked him, but he wasn't ready.