The case involves Dawnmarie Souza, who had to prepare a response to a customer’s complaint about her work. Ms. Souza, the board said, was unhappy that her supervisor would not let a representative of the Teamsters, the union representing the company’s workers, help prepare her response.

Ms. Souza then mocked her supervisor on Facebook, using several vulgarities to ridicule him, according to Jonathan Kreisberg, director of the board’s Hartford office, which filed the complaint. He also said she had written, “love how the company allows a 17 to become a supervisor”  17 is the company’s lingo for a psychiatric patient.

The labor board said that her comments “drew supportive responses from her co-workers” and led to further negative comments about the supervisor. Mr. Kreisberg said: “You’re allowed to talk about your supervisor with your co-workers. You’re allowed to communicate the concerns and criticisms you have. The only difference in this case is she did it on Facebook and did it on her own time and her own computer.”

An administrative law judge is scheduled to begin hearing the case on Jan. 25. Marshall B. Babson, a member of the National Labor Relations Board in the 1980s, said a broad company rule that says one cannot make disparaging comments about supervisors is clearly illegal under labor law. But he said an employee’s criticizing a company or supervisor on Facebook was not necessarily protected activity.

“There will arguably be cases where it is not concerted activity,” Mr. Babson said, suggesting that if a worker lashed out in a post against a supervisor but was not communicating with co-workers, that type of comment might not be protected.

If the Facebook conversation involves several co-workers, however, it is far more likely to be viewed as “concerted protected activity,” he said.

But employees might cross the line into unprotected territory if they disparage supervisors over something unrelated to work  for instance, a supervisor’s sexual performance  or if their statements are disloyal.

Courts often view workers’ statements as disloyal when they are defamatory and are not supported by facts. Mr. Babson cited a case upholding the firing of airline workers who held signs saying their airline was unsafe. But, he said, if employees held signs accurately saying their airline or restaurant had been cited for dozens of safety violations, that would most likely be protected.