No, high schools usually don't count unless they participate in some kind of program with Apple - and some do, even if their students don't know about it, so it can be worth investigating.
I was able to get a student discount just by picking a school from a list of colleges and checking a box that says I super double triple swear that I'm buying this product for an eligible student. And that was it. You used to need to supply a valid @.edu e-mail address.