Probably not, but, at this point, he’s been ruling by executive order, and people just listen to him. Its only illegal if the law is enforced. Otherwise, the law isn’t worth the paper its written on.

I don’t know the specifics of how everything works, but I could see a path where he could do that, legality of his actions notwithstanding.

My guess is that international students applying for student visas need to list what university they’re going to attend. I could see Trump ordering that the State Department simply decline any visas for students that are heading to Harvard. That would essentially accomplish Trump’s goals and leave Harvard in a position where they can’t really do much about it.