Can Texas secede from the union?
Legally Texas renounced the right to secede in specific terms, nothing vague about it:
March 15, 1866
Be it ordained by the people of Texas in Convention assembled, That we acknowledge the supremacy of the Constitution of the United States, and the laws passed in pursuance thereof; and that an Ordinance adopted by a former Convention of the people of Texas on the 1st day of February, A.D. 1861, entitled "An Ordinance to Dissolve the Union between the State of Texas and the other States, united under the compact styled 'Constitution of the United States of America,'" be and the same is hereby declared null and void; and the right heretofore claimed by the State of Texas to secede from the Union, is hereby distinctly renounced. Passed 15th March, 1866. The US Constitution is silent on the issue of secession. There is no provision in the Texas Constitution (current or former) that reserves the right of secession, but it does state that "Texas is a free and independent State, subject only to the Constitution of the United States" ... not to the President of the US or even the Congress of the US.
Both original and current Texas Constitutions state that political power is inherent in the people and (just as the Declaration of Independence declares) "the people have the right to alter their government in such manner as they might think proper."
Texas and Hawaii are two states that were once recognized as independent nations, before choosing to join the Union. Their voluntary decision to join the Union did not come with an explicit agreement that they could never leave.
Some people claim that the Civil War proved that secession is illegal. Whether one was in favor of the North or the South, all that war actually "proved" is that a state or group of states can be militarily forced to continue being a part of a group. Superior strength does not prove morality or legality as any citizen of the former Soviet Union can attest.
Some people are under the mistaken impression that the US Supreme Court decision in Texas v. White "proved" that secession is unconstitutional. Actually, that decision was not based on any precedent or anything in the Constitution and was in direct conflict with the actions of the then-President Grant who had to sign an act to "re-admit" Texas into the Union and allow them to send Representatives back to Congress. If Texas had never left, as the Court declared, it would not have been required to be "re-admitted" and Grant would not have needed to sign the declaration. This is a conflict that has never been fully cleared up.
Bottom line: There is no law forbidding or allowing secession. If Texas or any other state decides to secede, the resulting peaceful separation or war will depend not on law, but on the will of whomever happens to be Commander-in-Chief at the time.
One can also argue, and constitutional scholars certainly have, that the 'readmission' of Texas to the union did not violate the Supreme Court's decision in Texas vs. White...it was is superfluous, and indeed, did not, in and of itself violate, and was not contrary, to the Texas vs. White ruling. Simply speaking, there was no precedent for handling this situation. The readmission of Texas in early 1870 came just a short few months after Texas vs. White, and Congress and the President did not forsee the long-term implications of the Supreme Court decision. Just like there is no explicit wording in the Constitution forbidding secession, there is no wording in the Constitution specifically outlining the statutory process for the readmission of states.
Because of this, if state's rights proponents continue to argue that a state's right to secede is implied because they joined the union as 'independent states', a strong argument can also be made that an indissoulable union was also implied in the Constitution. Membership in a union does not dimish in importance or totality a state's sovereignity. That's the whole point of federalism. However, it does imply relinquishing some power to the national government. Indeed, federalism is 'shared' power, but it is not 'equal' power, at least in the example of the US. Clearly national dominance was designed into the structure of the Constitution.
Reality-based perspective: In truth, the US Constitution is not silent on the issue of secession. In Article I, Section 8, Congress is explicitly given the power to "suppress insurrections," and in Article I, Section 10, the states are specifically prohibited from "enter[ing] into any treaty, alliance, or confederation." Section 3 of the Fourteenth Amendment also makes it unambiguously clear that any person holding any elected office or military position who participates in "insurrection or rebellion" against the United States, has violated his or her oath and is permanently barred from holding office, specifically because of the crime of insurrection.
So, NO, Texas CANNOT legally secede from the Union.