When the government is an employer, it's wearing two hats: government-as-your-government and government-as-your-employer. The government-as-employer can punish employees for things it couldn't punish them for acting as government-as-government. Which things? It's complicated.
Government employee speech that is part of official duties is generally not protected. If your speech is part of your job duties, you can be fired and disciplined for it, and that's not a violation of your First Amendment rights. So, for instance, if you're a government public affairs officer, you can be fired or disciplined for what you say to the press or public as part of your job. If it's your job to run the government agency Twitter account, you can be fired or disciplined for what you put on it, and that's not a violation of your First Amendment rights. There are emerging exceptions to this rule — things professors say in their academic capacity are probably not covered by the rule — but those exceptions will probably be limited to positions that traditionally involve a high level of freedom from interference, like public university professors.
Government employee speech must be about matters of public interest to be protected: Public employee speech is only protected by the First Amendment when it is on a matter of public concern. That's a broad category, but doesn't cover merely internal matters. "This agency is destroying records the law requires it to maintain" is of public interest, "Bob in accounting is a suck-up and the Deputy Assistant Administrator lets him get away with murder" is not.
Public employee speech is only protected when the employee's interest in free speech outweighs the government employer's interest in discipline and order. If an employee is speaking on a matter of public interest and not in the course of their job, then their speech will be protected if the employee's interest in speaking outweighs the employer's interest in efficient and orderly operation. There's no bright line here. But in general, an employee's speech is most likely to be protected if it's on the employee's own time, on the employee's own platform or a platform not run by the employer, involves policy issues rather than personal attacks on people in the government workplace, and the employer can't show evidence of disruption of order or function. A letter to the editor of the local paper criticizing a government agency's policy choices is probably at the peak level of protection; an in-person confrontation in the office promoting factional disruption is probably at minimum protection.
The First Amendment isn't the only issue in play. Civil service regulations and statutes may have an impact on the ability to fire some government employees.
The First Amendment still protects government employees in their capacity as citizens. The government-as-employer can fire you for speech that would be protected if you were a private citizen, but can't, for instance, jail you for it. In other words, you still enjoy full First Amendment protection from government-as-government consequences, just not from government-as-employer consequences.
Edited to add: Speech by members of the military is completely different and not covered by this post.
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