This question was recently answered by the U.S. Supreme Court in a 5-4 decision. In the case of Garcetti v. Ceballos, the Court decided that employees must be acting as private citizens to receive First Amendment protection. That is, they are not acting as private citizens while performing their job duties. In essence, the boss can control what is said on the job, but not what the employee says on matters of public concern as a private citizen.
The Court stated, We hold that when public employees make statements pursuant to their official duties, the employees are not speaking as citizens for First Amendment purposes, and the Constitution does not insulate their communications from employer discipline.
So, to receive First Amendment protection, the employee must apparently be on his or her own time (clocked-out), and speaking in a forum for citizen expression on matters of public concern. Those employees who claim that they were just doing their jobs too well will find that they no longer have First Amendment protection. (They may still have protection under specific laws; for example, if they report environmental violations as required by federal law, then the environmental laws will still provide protection against adverse employment actions.)
The Garcetti decision is a setback for public employees, and we can look forward to further developments in the law as legislators consider the need for new laws, and courts issue further decisions in this area.