as for the branding, you can say no without conceding either side. the target of this twitterer seems to have done it successfully, keeping their job and not conceding either way.
note that i'm not taking sides here either. just making a point about having the fortitude to put social media in its proper place.
further, if you can't take a principled stand under pressure (another useful skill), it might be an indication that the stand isn't principled, or at the very least, you need to find the foundational principles on which to stand.