Can players on Steam OS play with people on PCs?

I'm thinking of getting him a Steam Box

Your plan won't work because there is no longer any Steam Machine hardware for sale. Most manufacturers discontinued or stepped away from their Steam Machine hardware by late 2016; at this point, I don't think anyone is building hardware with the specific intention of making it run SteamOS anymore, and it's not clear if Valve has any plans with it either.

Regardless, as others have mentioned, PlanetSide 2 is not available for Linux (which is what SteamOS is), and the developers explicitly stated in 2012 that they didn't plan on a Linux release. Given that PlanetSide 2 is now nearly 7 years old (and that the Mac port promised in that article never materialized), it seems highly unlikely that a Linux port would be added.