You're probably going to be fairly unlikely to find a boarding school with game-specific elements, though you should be able to find at least a few which offer programs for the arts (if you're interested in graphics, writing or design) and some which offer at least introductory computer science classes.

What you really want to focus on, though, is a school which provides strong fundamentals that will prepare you for your post-secondary studies. I, for example, went to a small, public high school in a rural area -- I was already interested in becoming a programmer, and specifically a games programmer. My school didn't really have anything in the way of CS courses -- we had word processing, some basic HTML class, an "advanced" computers class which was largely self-directed and reviewed/graded by the instructor and eventually a 2-year course which would earn students a Cisco Networking certification -- I ended up taking each of these because they were either required or they had some value.

Probably more importantly though, I took all the advanced math classes that were available -- I even took my college-level algebra course twice, because I ran out of math classes to take and just wanted to keep those skills fresh. By the time I had finished high school, I had exposure to geometry, calculus, statistics and college credit (through the programs at my high school) in algebra and physics. I also spent a year working as an assistant to the district computer/network administrator for an hour each day. All of this was valuable in my college career and the time since. The only classes I really regret not taking in high school ended up being chemistry (would have been nice to have more varied science education) and introductory auto mechanics (would have been nice to know a little more about cars -- having since discovered how much money you can save doing some stuff yourself.)