Should a police officer be fired, if they express views prejudicial to people that belong to minorities they are meant to serve, even if they express these views while off-duty?

Don't you think you're walking a bit of an edge there, minimising racism by equating it with criticism of an organisation? I hope you're aware the history of it isn't something that should be trivialised?

This case would be argued along exactly the same lines before the court had the student in question thrown in a racial slur. It will also have an impact on the ability for schools to punish students for racist cyberbullying in future. So I don't think it's problematic in the slightest to bring race into the equation.

One is criticism of the running and operation of an organisation; the other is steeped in the history of genocide and slavery, there are even laws in place around the world to protect people from it because of it's heinousness. But you're professing that advocacy of racism is no more malignant than basic complaints about a company?

I'm not sure if what you're advocating is because of perhaps missing education on the subject but this is a serious hot button issue at the moment; I'm honestly surprised you'd espouse this so freely.

An unreformed racist sprouting out bigotry and a teenage girl having a bit of a moan about the cheerleading squad are both protected forms of speech under the first amendment.

If the Supreme Court made an exceptionally broad ruling to the effect of forbidding any organ of the state revoking an appointment on the basis of first amendment protected speech uttered while not performing duties, then the ruling would protect a racist police officer and irate cheerleader equally.

The Supreme Court is, of course, very unlikely to make such a broad ruling in this case, but it does seem like that's the remedy which quite a few people in this thread would like to see decided.

As a matter of their employment agreement, sure.

I guess students sign behavior pledges or similar, but I'm more inclined to enforce an employment agreement against an adult than I am to give local school administrators complete control over what activities students they disapprove of participate in.

I believe as public employees, they should be held to at least the same standard as people in private employment. If I, as the employee of a company, make a Facebook post outside of work denigrating a group of people and my employer is made aware of that, I believe that I can be terminated. Should termination be the only solution? I don't think so. In a functioning society, there must be room for genuine contrition and redemption of those who have erred. I don't know what the proper method of addressing it should be, however.. The removal of the offensive speech and an immediate and heartfelt apology would be a firm minimum IMHO. The same should be true for public employees with the difference being that the appearance of impropriety should weigh towards sanctions. In both cases, pending conflict with employment contracts or local law, suspension, re-training, and demotions are all possible alternative sanctions.