Again, kinda sorta.
During working hours, you have no Constitutional "Freedom of Speech" beyond what the employer allows.
After all, as an employee during working hours you are representing the business, regardless of your position.
And courts have also ruled that your "after hours" activities may also be limited by corporate policies.
Even before social media, people could be disciplined for violating corporate code of conduct during their off hours.
(this is why people should NEVER "friend" their coworkers or employers on social media. Or at least create a different media page for coworkers and employers.)
It's all up to the company policies.
Again, that's why I tell people NEVER EVER EVER to allow your work and/or coworkers on your social media list.
Some employers are now "requiring" employees to allow the company to monitor their social media.
There is no way they can legally enforce it, unless it is in a contract. But that doesn't stop them from asking - and some people are dumb enough to let them.
Yeah ... that's totally stupid.
Like asking a criminal if they've committed a crime.
A criminal would likely have a "clean" social media and segmented drive they could "give up".