Can someone be fired from a job from writing a facebook post that does not put a name by directing it to someone?

Texas is an at-will employment state.  This means that an employee can be fired for any reason or no reason.  Employers don't even have to tell you exactly why.  The only prohibition on termination is that it cannot be based on discrimination related to a protected class (for example minority, female, etc.)  As long as the termination is not based on one of those reasons, then yes the employer can fire someone for a post on facebook.  The argument against termination is that you never really know who typed the words that were offensive to the employer.  However, an employer is free to reject or accept an employee's explanation for the appearance of comments.  Even if the employer only suspects that an employee is responsible for an offensive post, then they can proceed to termination... again... because Texas at will employment.  At-will seems harsh, but there are some advantages.  If an employee is fired because of the an unspecified post, then it's not really related to their job function.  This means the employee can file for unemployment and the employer will be hit with the costs of the claim.