The 'authority' mentioned was an employer, who is free to hire and fire people at will in he US, in many cases. Turns out that employing, say, a nazi, is not popular at many companies.

This can certainly go to far, but it's kind of up to the employer, isn't it.

I misread. You're right, it's fully reasonable for companies to remove individuals from employment due to their own speech (but nobody else's).