Can an employer of a private agency fire an employee based on Facebook activity?

No.  With the exception of the prohibition against slavery, the Constitution only prohibits government action, not action by private parties.  Moreover, while the First Amendment may give you an absolute right to say something, it usually doesn't shield you from the consequences of what you have said - for example, you can be sued for libel or slander if you make a false and defamatory statement about someone. While this is not a matter of Constitutional law, that doesn't mean that some other, lesser, statute might not protect you.   For example, many jurisdictions have "whistleblower" statutes which protect employees from being fired for disclosing serious wrongdoing by their employers.  If  the expired product was sold by your employer, and there was a serious health risk inherent in selling it, Georgia might have a whistleblower statute which applied (although it would be more likely if you were fired for reporting the circumstances to a government health agency rather than Facebook.)  In general, however, absent a statute (i.e. statutes which prohibit firing bases on race) or a contract providing otherwise (such as an employment agreement defining the only reasons you could be fired; sometimes an employee handbook can be sufficient for this purpose) , employees can be fired for any reason.