For one thing, you are stepping out of the range of free speech.
However we are still talking about institutional property.
It's still public property. Institutional or not has nothing to do with it. The question is whether the place or area is a public forum, or a limited public forum. Cougar high as a whole is a limited public forum, while there are areas on campus that are public fora. It's impossible to get away from that.
Seeing as how UH provided alternative venues, then it seems that they are not blocking free speech, but redirecting it to appropriate locations.
Which is legal, but a public university shouldn't use this ability in anything but a careful manner. At A&M this spring, they had some sort of idiocy called the Vagina chronicles or something stupid like that. They were allowed to display their message in A&M's public speech area. I didn't like it, but it appears that they are in good constitutional standing.