Flag protests and public employees

The assumption among supporters of protesting NFL players and critics of the NFL is that the league is trampling on the players' free-speech rights, that the players have a free-speech right to protest the anthem, save for the absence of state action. But the assumption is that if there were state action, the First Amendment would protect the players. Let's push on that question, with a hypothetical to which I genuinely do not know the answer:

The head of a government agency or office (it does not matter what level of government or what office) has decreed that the workday shall begin every day at 8:30 a.m. by everyone in the office standing before the flag with hands over hearts, recite the Pledge of Allegiance, and sing America, the Beautiful. The director explains that this symbolic reaffirmation of America reminds public officers of their obligations to the Constitution and to the public they serve in performing their jobs. Must an objecting employer, who believes that America's criminal-justice policies are discriminatory, participate in this ritual?

There are several doctrinal paths competing for attention here.

1) Barnette says students cannot be made to participate in the flag salute. By extension, it should mean other people cannot be compelled to participate in other patriotic rituals. Certainly Jackson's rhetoric speaks of patriotic rituals, not only the Pledge in schools. There also is a nice question of how far the Barnette protection extends--to speaking the words of the Pledge or anthem or to all engagement in the ritual. In other words, does Barnette mean you can opt-out entirely by kneeling or sitting or leaving the room? Or does it only mean you cannot be compelled to utter the word, but can be made to stand there, even at attention?

2) Employee speech rights within the workplace are limited, under the Garcetti/Connick/Pickering line of cases. Workplace speech that is part of the job is per se unprotected, while Connick/Pickering ask whether speech (whether in or out of the workplace) is on a matter of public concern and whether the employer's interests outweigh the employee's expressive interests. But on-the-job core political speech, however offensive, that does not affect government operations is protected. Thus a deputy sheriff could not be fired for stating, in a conversation with co-workers, her hope that a second assassination attempt on President Reagan would succeed.

3) Janus can be read to accord public employees greater protection against compelled speech than they enjoy against restrictions on their own speech, a criticism Justice Kagan leveled in her dissent. Kagan also predicted that Janus was about limiting public unions, not compelled speech generally, so a rule compelling employees to speak in a way other than donating money to a union.

So what might be the answer to my hypo? There are a couple of threshold question. First is how we should understand what the protesting employee (or an NFL player) is doing. Is he seeking to opt out of having to utter the government's message? Or is he trying to make his own affirmative statement about something (e.g., police violence)? This makes a difference between whether we are in Barnette/Janus or Garcetti/Pickering. Second is how much deference the court owes the government in defining what speech is part of the job. So will the court buy the government argument that the pre-opening patriotic ritual is designed to remind employees of their public duties and obligations and thus part of their public jobs.  And, if not and we are in Connick/Pickering, how disruptive of the workplace the court deems non-participation to be. Third, if this is compelled speech, can it really be that children in school enjoy greater protection against compelled speech than adults in the workplace?

Again, I do not know the answers, although I know I believe it should come out. Thoughts?