Texas public employers have broad rights to prohibit certain kinds of speech in the workplace, but those rights aren't unfettered.
If a government employee is speaking "as a citizen addressing a matter of public concern," the First Amendment requires that the government's interest in restricting speech be balanced against the employee's free-speech rights. But if an employer's rules only minimally restrict employees' speech, as in the following case, it's more likely to withstand First Amendment scrutiny.
Recent case: Urbano Herrera, a carpenter at the Medical Center Hospital in Odessa, was asked by his union to wear a "Union Yes" button on his uniform. Herrera did, but his supervisor told him to take it off because the hospital's dress policy required all employees to wear a uniform devoid of buttons. The only exceptions: pins showing one's education level (such as RN pins) or pins worn during annual events, such as the Great American Smoke-Out.
When Herrera refused to take off the button, the hospital suspended him without pay for three days. He sued, alleging a free-speech violation. The 5th Circuit Court of Appeals tossed out his case, reasoning that the restriction was a reasonable one and was enforced fairly. For example, employees also couldn't wear "Union No" buttons. (Communications Workers of America v. Ector County Hospital District, No. 03-50230, 5th Cir., 2006)
Final tip: Be sure to have a good reason for your dress policy and don't allow other buttons if you ban some.