Is a former employer in Texas allowed to release information about why the employee left or was fired?

"Allowed" is a tricky word.  There is no absolute prohibition in Texas law that prevents an employer from releasing such information and there is even a good faith protection for employers if they release information about a former employee to a prospective employer which the former employer reasonably believes to be true. That said, realeasing such information is fraught with risk, such as defamation or retaliation suits from former employees if the information is untrue or if the employer gave out the information for the purpose of retaliating against an employee for some exercise of the employee's rights.  Most employers limit their disclosures about former employees to confirmation of employment, dates of employment and last position or title and refuse to say why the relationship terminated.  Good luck.