Regarding whether a public employee’s rights are protected under the First Amendment, the Supreme Court has “made clear that public employees do not surrender all their First Amendment rights by reason of their employment. Rather, the First Amendment protects a public employee’s right, in certain circumstances, to speak as a citizen addressing matters of public concern.” This is a quote from Garcetti v. Ceballos, 547 U.S. 410 (2006).
The Supreme Court also acknowledged in Pickering v. Board of Education (1968) that “the threat of dismissal from public employment… is a potent means of inhibiting speech.”
These examples show that the Supreme Court has afforded public employees a substantial, although not unlimited, amount of protection under First Amendment rights.