Q: Do private school students have any recourse when their free speech rights are stripped?
A: Options for fighting private school censorship can vary significantly from school to school. As you are probably already aware, you don't have First Amendment protection from censorship in private schools, so you have to look elsewhere. Reminder: The First Amendment only protects against censorship by government officials; it would not restrict censorship by, for example, the headmaster of a private school. If the school is in either CA or RI we'd first look at possible state statutory protection. (Other states may join the lineup as time goes by.)
Otherwise, you're largely dependent on school policies for protection. While private schools don't have to promise their students that they will respect their free speech rights, some (in fact, many) schools have voluntarily adopted policies/statements that do so. Where that's the case — if the promise is specific enough — courts have looked at such documents, which students reasonably rely upon, as creating legally enforceable contractual promises. Look, for example, for a Student Rights and Responsibilities statement or a specific student media policy. A few states also have state constitutions that may provide additional free speech protections in the private school context. Check out the SPLC Private School Guide for a deeper dive into these topics.
Where they can’t rely on a court of law for protection, many private school students have forced schools to rethink their censorship by taking their case to the court of public opinion (local media or other public relations outlets), and holding administrators accountable for their actions. Thankfully, in most cases, censorship is a bad look — and lousy PR — for schools.