In: Operations Management
Yes they do.
Companies have governance systems with established policies and procedures. An employee working with a company is the face of that company and whatever he says at work or off work, is unofficially, a statement made by the company. They can regulate what you speak about the company, about political, cultural and other factors concerning the industry etc. Recently, an Indian tech giant had fired an employee for tweeting inappropriate content regarding the corona virus pandemic. If you are working with government, these things will be explicitly stated in your contract papers itself. Some companies even have regulations regarding the loudness of your speech.
Some countries have laws made like if two or more employees want to speak something inappropriate regarding the working conditions be said about a company, they can do so under the particular clause. But again, loose talk is not covered under such laws. If an employer's actions affect a group of people and if they want to bring to the notice of public or the law, they are permitted to do so. If there are contraditions between the state's laws and company's policies, then the laws of the state will prevail. But even with all these, if your actions cause any serious damage to business, comapnies can sue you. Afterall who wants their life to become miserable?