Your right to freedom of speech as an individual is separate from your right to expression in a public role.
Public employees and private employees are treated differently under these laws. While public-sector employees retain their right to free speech while employed by their government, private-sector employees generally do not have that enforceable right against their private employers because constitutional rights and the right to free speech only protect us from the government.
The Supreme Court has ruled that public employees do not relinquish their First Amendment rights by entering into employment with a government entity—as long as they speak as a private citizen regarding a matter of public concern.
But if your public-sector role makes you a high-level policymaker or requires the speech as part of your job, your constitutional right to free speech would not give you job protection.