The Volokh Conspiracy
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That turns out to depend on the state where the employee is employed. (Note that I'm speaking here of firing for attending such a rally or speaking at it, not engaging in criminal violence connected with it.)
1. The First Amendment applies only to government employers; it doesn't apply to nongovernmental entities (whether or not those entities have government funding or contracts). Let's focus then on private employers who fire such employees based on their own judgment (not because of some pressure from the government, which may well be unconstitutional).
2. Private employers are, of course, bound by various statutes. But federal employment law bans discrimination based on race, religion, sex, national origin, age, disability and various kinds of labor-union-related activity. It doesn't ban discrimination based on political affiliation.
3. A substantial minority of states, though, do ban discrimination based on political activity, especially off-the-job political activity; some cities and counties do as well. I cataloged them in 2012 in my article "Private Employees' Speech and Political Activity: Statutory Protection Against Employer Retaliation"; since then, Utah has enacted a similar statute as well.
a. Some statutes ban employers from firing employees for "political activity," including ideological advocacy generally and not just election-related politics. California, Colorado, Louisiana, Minnesota, Missouri, Nebraska, Nevada, South Carolina, Utah and West Virginia seem to fall in this category; there are similar ordinances in Seattle and Madison, and a New Mexico statute may also fit here, though it's a bit more ambiguous.
b. Connecticut protects employees from retaliation for their speech more broadly.
c. Colorado and North Dakota ban employers from firing employees for any off-duty lawful activity; that would cover speech as well.
d. New York bans employers from firing employees for off-duty "recreational activities"; it's not clear to what extent going to rallies would qualify.
e. Still other statutes apply to belonging to, endorsing or affiliating with a political party, something that probably wouldn't cover ideological rallies such as this—we see that in D.C., Iowa, Puerto Rico, the Virgin Islands, Broward County (Fla.), and Urbana (Ill.).
f. Illinois, New York and Washington laws apply to election-related activities, which again wouldn't apply just to rallies. (In New York, this is in addition to the recreational activity statute; some states, such as New York, have two statutes that protect two different kinds of activities.)
g. Some states have laws limited to signing election-related petitions, or giving campaign contributions: Arizona, D.C., Georgia, Iowa, Minnesota, Missouri, Ohio, Oregon, Washington, and (the campaign contributions laws) Louisiana, Massachusetts, and Oregon; Hawaii, Idaho, Kentucky, Tennessee, West Virginia, Wyoming, and Guam might also fit this category.
4. Some of these laws actually make it a crime to fire an employee on these grounds; I think all would likely authorize civil lawsuits.
5. Some apply to refusals to hire as well as to firing; others don't.
6. What if the employer concludes that the off-the-job speech has badly hurt morale or relationships with customers? Some of the statutes expressly provide that the employer has some latitude in such cases (though it varies statute by statute, and such exemptions are generally fairly narrow).
Others seem to protect all political activity, even highly controversial activity. The Louisiana Court of Appeal, for instance, has held that the ban applies even when "the 'business' justification for firing plaintiff in this case is a real one," such as that plaintiff's political advocacy "would antagonize persons who could withdraw business from plaintiff's employer." And this is consistent with other antidiscrimination laws: For instance, an employer can't fire an employee based on the employee's religion even when coworkers or customers very much disapprove of that religion, and threaten to quit or boycott the company.
7. In principle, such state laws could be preempted by federal law, and I've heard some suggest that such firings might be called for by workplace harassment law, on the theory that the very presence of a known white supremacist or neo-Nazi employee would create a "hostile work environment" for nonwhite or Jewish employees. But I don't think that would be a viable defense for the employer.
I do think that hostile environment harassment law can sometimes pose serious First Amendment problems (because it involves the government pressuring private employers to restrict speech in the workplace based on its content and viewpoint); and some, though by no means all, of the few courts that have considered the issue have likewise said that the First Amendment may preempt hostile environment harassment claims in some situations. But whatever one thinks of that debate, I've never seen any case that holds employers liable for a supposedly hostile environment created by an employee's off-the-job political speech (at least when the speech isn't specifically targeted at the employee's particular coworkers). There's thus no conflict between federal harassment law and these state speech protection laws, at least when it comes to off-the-job speech.
UPDATE: Some commenters noted that most employment is "at-will employment," under which an employer can fire an employee for any reason, so long as the reason isn't statutorily forbidden. That is true. (When someone isn't an at-will employee, for instance because he has a union contract or a tenure contract, then there would also be the question whether the contract allows such firing.) But the whole point of the statutes is that they do forbid a particular reason for firing someone—that person's political activity. Just as bans on discrimination based on religion, race, sex, or (in those states that so provide) sexual orientation are limits on at-will employment, so are bans on firing people based on their political activity.