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There is no general constitutional right of employers to refuse to hire people whose statements or actions are "profoundly in conflict with [the employer's] values as a firm." Federal and state laws, for instance, generally forbid discrimination in employment based on the employee's religion, however much the religion might profoundly conflict with the employer's values.
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Likewise, a considerable number of states and some counties, cities, and territories ban discrimination based on various kinds of political activity, or speech more broadly (though a considerable number don't). The California statutes protecting private employees from retaliation for their "political activity," for instance, have been read as banning discrimination in firing or hiring based on an employee's "espousal of a candidate or a cause," including broad ideological causes and not just ballot measures.
Other laws are narrower, or sometimes vaguer. New York law, for instance, bans employment discrimination based on off-duty "political activities," defined to mean "(i) running for public office, (ii) campaigning for a candidate for public office, or (iii) participating in fund-raising activities for the benefit of a candidate, political party or political advocacy group." That seems largely limited to election-related speech, which wouldn't include advocacy of Hamas's actions or of Israel's retaliation for those actions.
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Now what if the employer argues that the employee's or prospective employee's speech is so offensive to clients or coworkers that it undermines the employer's business? The laws also appear to vary as to that. Recall that, for instance, an employer can't raise such objections as a defense to firing employees based on their religious values; however much your customers may disapprove of Satanists or evangelical Christians or Orthodox Jews or Sunni Muslims, you can't use that as a basis for rejecting the employee.
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Again, not all jurisdictions have such laws. But some do, and those may well provide legal protection for employees' political speech, including speech that many view as highly offensive. ...
May Private Employers Fire (or Refuse to Hire) Employees Because of Their Praise of Hamas (or Praise of Israel)?
In recent days, we've seen some employers saying they'd refuse to hire people based on those people's praise of the...
A more in depth dive on the issue can be found here (I'm quoting the conclusion, the essay covers the issues in more depth if you are interested):
As I hope I’ve made clear, laws limiting private-employer-imposed speech restrictions might be a cure that’s worse than the disease. They might unduly interfere with employers’ associational rights (though not generally in an unconstitutional way). They might unfairly require employers to keep paying employees who are more trouble than they are worth. They might make it harder for employers to dismiss employees even for eminently legitimate reasons unrelated to the employee speech. And of course, they might increase the amount of offensive and harmful speech by making such speech less costly for the speakers (though the same can be said of free speech rights generally).
At the same time, private employer speech restrictions genuinely do threaten to undermine democratic self-government, the marketplace of ideas, self-expression, and the development of autonomous citizens, much like many governmental speech restrictions do. To take just one example, consider abortion, which the Supreme Court has now returned to the political process. It’s much harder to have meaningful democratic debate about this subject if people know that they can be fired for signing an initiative or referendum petition, or for contributing their money to a ballot-measure campaign, or for publicly endorsing a candidate—or, for that matter, just for expressing their views on the moral or practical impact of one or another position. And to the extent that religious association or expression on this subject is protected by Title VII and state religious discrimination bans, nonreligious expression ought to be as well.
This threat to public discussion and to self-expression also seems likely to be increasing, as people find it easier than ever before to demand, in an organized way, the firing of other people whose speech they condemn. And the existence of such laws may take some of this public pressure off employers, by giving employers an answer to such demands: “We don’t like the employee’s speech, either, but we can’t fire him, because the law has tied our hands.” I’m not sure what the right answer is to these questions; but I hope what I’ve said above can help us think through them.