Yes.  Some employers can penalize employees for using social media. Here are some reasons to be careful about your social media activity:

Your Constitutional Right to Free Speech May Not Apply.  Some believe social media activity is protected as a form of free speech under the First Amendment of the United States Constitution.  However, this right only protects individuals from unreasonable restriction of speech by the government.  This right does not apply to actions taken by private employers that limit or penalize employee free expression such as social media activity.  If your employer is not a governmental entity, it is very possible that you could be lawfully terminated because of what you post on social media with some narrow exceptions.

Other Protections are Limited.  Some laws prohibit employers from penalizing employees for their speech (including social media activity), but these laws are narrow. The National Labor Relations Act protects the rights of employees to discuss wages, benefits, terms and conditions of employment, unlawful employer activity, and joining or supporting a union, but it does not extend to other matters.  Other federal laws, such as Title VII of the Civil Rights Act, prohibit employers from using social media posts as a false reason to penalize an employee when the actual reason was due to unlawful discrimination.

The Past Can Haunt Your Future.  If you use social media, do so with caution.  Internet posts are like tattoos; they are visible forever and as such, future employers will use them to make judgements about you.