QUESTION

Can we adopt a policy banning employees from posting anything about work or the company on social media?

SHORT ANSWER
A broad ban on posting on social media is not supported by the law, and some employee postings are protected by the National Labor Relations Act.
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While an employer can have a policy that addresses using professional judgment about social media activity about the company, a broad ban is not supported by the law. Certain types of employee speech on social media is protected by the National Labor Relations Act (even if you are not a unionized work place) as it is considered speech related organizing and addressing working conditions.  For example, sharing information relating to rates of pay, overtime, scheduling and management disciplinary process will all fall squarely in protected activity.