Can a company even tell anyone that you were fired???

While this is incredibly unprofessional it is not against the law.  There is no right to privacy in such a situation.  Unless the remarks were untrue, you basically have no cause of action here.

Most employment relationships are what is known as "at will".  This means that you can work for an employer, or not, your choice.  In turn, the employer can hire or fire someone for any reason or no reason whatsoever.  Included in the latter is how you are fired as well.

The only exceptions to this are: 1) If there is an employment agreement or contract (which governs termination); 2) If there is a union or collective bargaining agreement (which governs termination); 3) If there is an employee handbook or other official policy statement (which governs termination); 4) If discrimination is a factor (there can be no termination for reasons of race, religion, age, disability, sex, national origin).