Q:
Doesn't the Texas Constitution reserve the right of Texas to secede?
A:
This heavily popularized bit of Texas folklore finds no corroboration where it counts: No such provision is found in the current Texas Constitution (adopted in 1876) or the terms of annexation. However, it does state (in Article 1, Section 1) that "Texas is a free and independent State, subject only to the Constitution of the United States..." (note that it does not state "...subject to the President of the United States..." or "...subject to the Congress of the United States..." or "...subject to the collective will of one or more of the other States...")
Neither the Texas Constitution, nor the Constitution of the united States, explicitly or implicitly disallows the secession of Texas (or any other "free and independent State") from the United States. Joining the "Union" was ever and always voluntary, rendering voluntary withdrawal an equally lawful and viable option (regardless of what any self-appointed academic, media, or government "experts"—including Abraham Lincoln himself—may have ever said).
Both the original (1836) and the current (1876) Texas Constitutions also state that "All political power is inherent in the people ... they have at all times the inalienable right to alter their government in such manner as they might think proper."
Likewise, each of the united States is "united" with the others explicitly on the principle that "governments derive their just powers from the consent of the governed" and "whenever any form of government becomes destructive to these ends [i.e., protecting life, liberty, and property], it is the right of the people to alter or to abolish it, and to institute new government" and "when a long train of abuses and usurpations...evinces a design to reduce them under absolute despotism, it is their right, it is their duty, to throw off such government, and to provide new guards for their future security."
Q:
Didn’t the outcome of the “Civil War” prove that secession is not an option for any State?
A:
No. It only proved that, when allowed to act outside his lawfully limited authority, a U.S. president is capable of unleashing horrendous violence against the lives, liberty, and property of those whom he pretends to serve. The Confederate States (including Texas) withdrew from the Union lawfully, civilly, and peacefully, after enduring several decades of excessive and inequitable federal tariffs (taxes) heavily prejudiced against Southern commerce. Refusing to recognize the Confederate secession, Lincoln called it a "rebellion" and a "threat" to "the government" (without ever explaining exactly how "the government" was "threatened" by a lawful, civil, and peaceful secession) and acted outside the lawfully defined scope of either the office of president or the U.S. government in general, to coerce the South back into subjugation to Northern control.
The South's rejoining the Union at the point of a bayonet in the late 1860s didn't prove secession is "not an option" or unlawful. It only affirmed that violent coercion can be used—even by governments (if unrestrained)—to rob men of their very lives, liberty, and property.
It bears repeating that the united States are "united" explicitly on the principle that "governments derive their just powers from the consent of the governed" and "whenever any form of government becomes destructive to these ends [i.e., protecting life, liberty, and property], it is the right of the people to alter or to abolish it, and to institute new government" and "when a long train of abuses and usurpations...evinces a design to reduce them under absolute despotism, it is their right, it is their duty, to throw off such government, and to provide new guards for their future security."
