Can they do that under the First Amendment? Eugene Volokh says no. The LA Times solicited views from a number of experts who unanimously concluded no (with one anonymous attorney calling it a "gray area"). And then there is Noah Feldman, who concluded that the university may expel the students because they face potential liability for allowing a "hostile educational environment." And, indeed, this appears to be the theory on which the university's president is relying.
I think everyone agrees that:
1. A public university may not expel students for speech protected by the First Amendment; and
2. Even horribly offensive, racist speech is protected by the First Amendment generally (sometimes the First Amendment yields uncomfortable results - the Westboro Baptist church has a right to picket funerals with signs like "Fags Doom Nations").
Strangely (especially for a Harvard law professor), Feldman never tells us what a "hostile educational environment" really is. He notes that the principle applies in Title IX (sex discrimination) . . . and it does. The Supreme Court defined the phrase as follows:
[A] plaintiff must establish sexual harassment of students that is so severe, pervasive, and objectively offensive, and that so undermines and detracts from the victims' educational experience, that the victim-students are effectively denied equal access to an institution's resources and opportunities.Davis Next Friend LaShonda D. v. Monroe Cnty. Bd. of Educ., 526 U.S. 629, 651, 119 S. Ct. 1661, 1675 (1999). I won't take that next step of analyzing whether frat bros singing on a private charter bus among themselves rises to this level, but I did want to fill this noticeable gap in Feldman's analysis.
If you are interested in this topic, the Third Circuit's opinion in Saxe v. State College Area School District addresses the "the very real tension between anti-harassment laws and the Constitution's guarantee of freedom of speech." The Court struck down a high school harassment policy as overly broad, holding that there is no "categorical 'harassment exception' from First Amendment protection." Two fun facts from that case: 1. I graduated from State College Area High School (attended grades 1-12 in that school district); 2. the opinion was written by now-Justice Alito.