No I wouldn't, people have the right to freedom of speech, even if we don't like it, that's how women get sacked or arrested for GC views. I would challenge it, but reporting something someone said on social media to their employers is over the top. (Although there's a reason I don't have my social media accounts linked to my job.) no, using racist language is not freedom of speech it is illegal.
I wish people would stop trotting out the “I believe in freedom of speech” crap to justify doing nothing.
And even if you call racist language “freedom of speech,” we also have freedom of action.
If someone ends up losing their job because of having broken the law then that is down to them,not me or whoever reported them. Again there’s far too much of this “What if they can’t feed their family” mentality out there which prevents people doing anything about others, such as reporting them for benefit fraud, or other theft, or racist language, etc etc etc.
Most companies actually have a social media policy now, so your employer as a rule does have the right to look at your social media, in fact access to your social media is often a part of pre employment screening for many companies.
If I was an employer I would dismiss them instantly, and I would have cast iron clauses in my employees handbook which would make it very clear that’s what would happen in the event of them e.g. spreading hate speech etc. And I would make it very clear that I would refuse them a reference.
I also think though that people need to call it, challenge it, but never share it, even to be outraged over it. Because the more people who share the more the message is spread, and not everyone who sees it from your share will be outraged about it, some will share it to their equally racist connections.