In my mind, there are some similarities here to the affirmative action conundrum.
As background, I'd say that sometimes in romantic relationships, there's a personality clash that makes people a poor fit for each other. Sometimes the same can happen in employment relationships. I'd never want to tell an employee that she wasn't allowed to quit her job because her employer had personal views she abhorred. If a relationship isn't working out for personality reasons (including incompatible political views), either party should be able to end it and move on, IMO.
Where this is similar to affirmative action is in the complicating distinction between private and public employers.
While I'd give a lot of leeway to private parties deciding whom they do and don't want to associate with, I'm extremely leery of public employers discriminating based on race or on expressions of political views.
I'm quite comfortable, for example, allowing private schools to have race-based affirmative action programs if they think such programs will benefit their students' educational experiences. State universities are a harder call because, on the one hand, government-sanctioned racial discrimination is icky due to its terrible history, but on the other hand, if state universities are going to compete with private universities, they should be able to adopt programs that benefit their students' educational experiences, same as private universities do.
By the same token, I'm quite comfortable allowing private actors to end employment relationships that aren't working out due to personality clashes if that will create a more harmonious work environment. But I'm way less comfortable when the government decides which views may or may not be expressed during its employees' time off from work.
The link above concerns a public-school principle, I believe. Terminating her for holding stupid views should depend, IMO, on whether the views in question are objectively stupid or merely subjectively stupid, which can be a fuzzy line. So it seems complicated.