How would Texas protect the rights of people who voted no?

The same way it protects the rights of everyone else: under the Texas Bill of Rights, which already guarantees them, and as full citizens of a Texas that belongs to all Texans, not only to those who voted yes.

Their rights are already written down

Texans do not depend on Washington for their fundamental freedoms. The Texas Constitution opens with a Bill of Rights that protects free speech, religion, due process, property, the right to keep and bear arms, and more, and in several respects it protects them more strongly than the federal Constitution does. Those protections are already in force, they apply to every Texan regardless of how they voted, and independence does not weaken a single one of them. If anything, an independent Texas answers to its own Bill of Rights directly, without it being second-guessed from outside.

A free vote includes the freedom to lose one

Independence is decided by a referendum, the most direct voice the people have. In every election, some people are on the losing side, and they remain full and equal citizens the next morning. That is how self-government works. A Texan who votes no is not a second-class anything. They are a Texan whose side did not carry a particular vote, exactly like a Texan whose candidate lost an election, and they keep every right, every freedom, and every claim on their government that they had the day before.

Independence is not a purge, it is a homecoming

There is no version of Texas independence that involves treating fellow Texans as enemies. The entire premise of the movement is government by the consent of the governed and closer to the people it serves. A government built on that premise cannot turn around and disenfranchise the very neighbors it claims to represent. People who voted no will vote in the next election, run for office, own property, speak freely, worship freely, and shape the country Texas becomes. Many who oppose independence beforehand become invested in making it succeed once it is the shared project of the whole state.

Your citizenship does not hinge on your ballot

Citizenship in an independent Texas would follow from living here legally, not from how anyone marked a referendum. A Texan who voted against independence is a citizen on exactly the same terms as one who voted for it. There is no asterisk, no loyalty test, and no penalty written into the result.

The bottom line

Texans who vote no keep every right they have, protected by a Texas Bill of Rights that already guarantees them, as equal citizens of a country that is theirs too. Self-government means governing for all of Texas, including the Texans who disagreed.