Employers have the right to hire or not hire whomever they want barring discrimination against protected classes, based on whatever criteria they want to consider relevant. "Person who said a stupid thing on the internet" is not a protected class.
A Twitter mob doesn't have any objective power to prevent anyone from finding employment. They do, however, have free speech rights which include the right to communicate with someone's potential employer and argue that they should not be hired. If literally every employer decides not to hire someone based solely on what that mob says, then (notwithstanding how ridiculous that scenario would be in real life) that's possibly a labor rights issue more than a free speech issue.
