It is complex.
Every private entity can 100% control speech on its own property.
However if this is a State school (I know it is UH, but, for example, the University of California system is more private than public) different rules com into play (including suppression of free speech for the workers who can’t wear campaign buttons, etc.).
I think this is a righteous suit, but it isn’t a legal slam dunk.
What do you think the reasoning would be to prevent students from passing out copies of the U.S. Constitution?
” ... the University of California system is more private than public”
The UC system is public, despite the fact that it receives less than half of its funding from the state. Restrictions of speech on public property are viewed very skeptically by the courts.
In this case, UH would have to argue compellingly that some important public purpose (safety, for example) is served by the restrictions.
Where is the 13th Amendment Zone?