It is a common misunderstanding that Texas has the right to leave the United States. The Texas State Library and Archives Commission (TSLAC) addresses this on their page about Texas’s annexation:

It is said of Texas (and, occasionally, Vermont) that it received a letter or document of permission to withdraw from the Federal Union if it so chose. […]

In fact, Texas received no special terms in its admission to the Union. Once Texas had agreed to join the Union, she never had the legal option of leaving, either before or after the Civil War.

This page links to two governmental acts about Texas and its statehood:

• The Presidential Proclamation declaring peace with Texas after the Civil War
• The 1870 Act readmitting Texas to the United States

According to TSLAC, neither document contains language that allows Texas to leave the United States.