Where a university's imprimatur is a prerequisite for a specific professional certification, the university is allowed deference in matters of regulation of speech by the candidate.
A candidate in a university professional certification program is not necessarily protected under the public employee speech doctrine.
A student teacher may be removed for conduct that, in the reasonable professional judgment of the university, violates defined and established professional standards which are narrowly tailored to serve the foundational mission of the program.
Programs can look to speech as an indication of likely future conduct.
Academic dismissals do not trigger due process interests so long as they are careful and deliberate.
https://d3bsvxk93brmko.cloudfront.net/datastore/opinions/2015/12/29/13-16524.pdf
Since 2011. An occasional commonplacing of the U.S. Circuit Courts of Appeal.
Neither an authoritative statement of the law nor legal advice. Quick work.