Well, there's one anecdote right upthread where Manmademan points out that in his government office the rule is basically "employees are not allowed to post anything controversial to social media". Government jobs have a pretty big problem with this in general. You get occasional stories about teachers getting fired because parents don't like what twenty-somethings do for fun. I would guess that a really large number of employees self-censor what they non-anonymously post online because of concerns about this, though I don't have numbers. But I don't think I need numbers to say that the sort of argument you actually get from liberals about free speech and employment is not very liberal, though them being unaware of a bigger problem could explain why they don't seem to have thought much about the issue.
As for how important this is, I think from a liberal perspective this just depends on how important you think free speech is. Obviously it's pretty easy to rhetorically downplay the importance of being able to say controversial things, but the arguments for why free speech is a good thing are old and easy to find and I don't really see the need to get into them. My argument here is that liberals don't have good reasons to distinguish in the way that they often do between the government suppressing speech and employers (including in some cases the government acting as an employer) suppressing speech.
I actually feel like modern liberalism tries to do quite a bit to maximize liberty, with some limits to just how much redistribution we're willing to do or how much we're willing to help poor people in other countries. The things you can't touch are the things where enforcing the law would itself be more of an imposition on liberty than what you're trying to correct. So, like, you could make an argument here that employers just have a really important freedom-of-association right to employ whoever they want, but this would not be very persuasive because in general liberals don't seem to feel like this is actually a really important feature of the employer-employee relationship - that's why liberals like anti-discrimination laws. On the other hand, it is bad that racists don't want to associate socially with the people they're bigoted against, but people's right to hang out with whoever they want is really important and we're not going to try to legislate and micromanage that people not be racists in deciding who they're friends with. What things do you have in mind?
This is probably the last from me for a while because I lost my home internet connection and won't have it back on for a little while.
