St Paul's Catholic School in Jax Beach is probably the most reputable private school in Jax Bch- however, I believe there are requirements about participation in the church's community(double check on that, though). Other notable schools are Bolles, and Episcopal- but they aren't in Jacksonville Beach. There are other private schools in Jax Bch but, they are smaller and I'd recommend interviewing with them to get a feel for what the school is like and if it's a good fit for you.
As far as public schools go, Fletcher Middle is one of the best around.
Hope that helps!