Facebook is now the third largest country in the world. That is, if each user of Facebook was a citizen of a country, then that country would be the third largest. Facebook, Myspace, Linked, the list of social media web sites grows longer each year. Blogging grows leaps and bounds every year. Cases in which employers have tried to constrain what its employers write on these sites continues to grow.
Employers have some risk if they try to control what an employee writes. If an employee posts information about his color, racial background, religion, disability, age or gender and those characteristics then lead to termination at work, the employee may have a right to pursue a discrimination claim.
Some states, not Texas, have off-duty conduct laws. Off-duty conduct laws provide that an employee may not be terminated for off-duty conduct that has no effect on the employer.
An employee might write something online in opposition to discrimination in the workplace. if the employer then takes some action against that employee, then any resulting adverse personnel action may constitute retaliation. Retaliation for opposition to discriminatory practices is prohibited by Title VII of the Civil Rights Act of 1964, and all other civil rights statutes.
If an employee complains online about "terms and conditions" of employment, then the employee is protected by the National Labor Relations Act. The NLRA was designed to allow unions to form. Typically unions start with complaints or discussions about workplace conditions. To qualify, such discussions must be "concerted" and must be for the "mutual aid and protection" of more than one employee. Concerted activity has always been protected. But, now it is simply protected in the new online venue.
That does not mean the employer cannot forbid employees from publicly disparaging their products or the products of competitors. In one NLRB (National Labor Relations Board) filing, the union started a facebook page. Sears objected to the extent that the website disparaged Sears products or the products of competitors. The NLRB essentially agreed.