They can continue to reside as US Citizens if they wish, or they can leave if they wish. Not much of a problem there if you ask me.
However, if enough Texans favored the idea, maybe they could convince the rest of the country to amend the Constitution to permit a "secession."
No need. The founding charter of the nation trumps Constitutional law. It is under the authority of this document and what it represents that the Constitution posses it's borrowed "authority."
If the right asserted in the Declaration is false, then the Constitution is Null and Void, because it is under this right that the nation lays it's own claim to independence.
"This Constitution, and the laws of the United States which shall be made in pursuance thereof; and all treaties made, or which shall be made, under the authority of the United States, shall be the supreme law of the land; and the judges in every state shall be bound thereby, anything in the Constitution or laws of any State to the contrary notwithstanding."
The citizens of the United States living in Texas are entitled to the privileges and immunities that flow from Article VI as well. If you want a new country, you need to find some new soil somewhere. And, as the Israelis can tell you, it's not easy nowadays.
But, again, the most important hurdle is that the people of Texas won't stand for it. Don't you believe that their feelings should count? Of course, you do.