Since many answers aren't actually answering the question...
Most places require a permit to assemble en masse. There are "free speech zones" where you can create large gatherings without any kind of advance notice or permit or whatever. Most universities have a free speech zone towards the middle of their campuses. Cities will also often have at least one but somewhere that doesn't inconvenience commerce, like a park or near city hall.
Most mass assembly requires a permit and sometimes a fee, even in public places. Following this prevents arrest by "disturbing the peace" or other such laws, usually.
How this squares with the first amendment is interpretation. Individual freedom of speech is protected except very specific public order and safety things, e.g. calling for violence. Coordinated, mass freedom of speech is perceived as a fast path to rioting.
I'm not saying this is right, but this is my understanding of how things work. I'm not a lawyer or an expert in these matters. This is just what I learned from activist friends in my university time ages ago.
As for expulsion, public universities are run by states. To my understand, Trump has no legal mechanism to do this. He's just talking out of his ass or expects to bully public institutions into expelling students by threatening to withhold department of education funding... but he's planning to kill that anyway, so 🤷♂️
I spent most of my 20s doing grassroots campaigning, with a large part of it being protest planning and organization. You generally do not need a permit to conduct protests on public property, even large ones, in the USA.
The exceptions are very specific; i.e. blocking roads may require a permit.
An illegal protest, according to conservatives, is any protest they disagree with. Waving picket signs or blocking traffic is disruptive, destructive, and illegal. Storming the capitol is, on the other hand, a perfectly acceptable peaceful protest.
Let the record reflect that, while this has not yet occurred, the fact that you have to ask is concerning enough to warrant treatment as if it has occurred. Furthermore, college campuses are one of the primary places that protest movements begin in recent years. Declaring protest illegal in nebulous terms like this is a HUGE swing toward breaking open the full fascism piñata.
He doesn’t give a shit about the speech. It’s the schools and universities he wants to harm. Even if it’s just a matter of disrupting them through legal overwhelm.
Remember, he’s only got the intelligence of a kumquat. He doesn’t even understand that he’s too stupid to understand anything. He vomits whatever he thinks give his followers happy endings.