Does Texas have any special legal claim because of how it was annexed?

Texas has a strong claim, and it rests on two pillars: the right several states reserved when they ratified the Constitution, and the doctrine that carries that right to every state admitted afterward, Texas included.

Three states reserved the right to take their powers back, and Congress accepted it

When the Constitution was being ratified, three states did not simply sign on. They ratified while putting on the record that the people retained the right to resume the powers they were delegating. Virginia, on June 26, 1788, declared that the powers granted "may be resumed by them whensoever the same shall be perverted to their injury or oppression." New York, on July 26, 1788, declared "that the Powers of Government may be reassumed by the People, whensoever it shall become necessary to their Happiness." Rhode Island used nearly identical reassumption language on May 29, 1790. These were not side letters. They were part of the formal instruments of ratification. And Congress seated all three states without objection. The reservation became part of the constitutional bargain.

Equal footing carries that right to Texas

Here is the bridge to Texas. Under the Equal Footing Doctrine, every state enters the union on completely equal terms with the original states. The Supreme Court has been emphatic about this. In Coyle v. Smith (1911), the Court described the union as one "of States equal in power, dignity and authority, each competent to exert that residuum of sovereignty not delegated to the United States," and struck down a condition Congress had tried to attach to a new state's admission. The Court in Pollard's Lessee v. Hagan (1845), the year Texas joined, held that new states enter with the same full sovereignty as the originals. So whatever rights Virginia, New York, and Rhode Island reserved, Texas holds too. You cannot have an original thirteen states with a reserved right to resume their powers and thirty-seven later states locked in without it. That would be a union of unequal members, which is exactly what equal footing forbids.

Madison confirmed the compact runs on consent and breach

The Framers understood the union as a compact among sovereign states, and a compact carries the ordinary rules of agreements. Madison said so directly. In Federalist No. 39 he described each state as "a sovereign body, independent of all others, and only to be bound by its own voluntary act." In Federalist No. 43 he applied the law of compacts to this very union: "a breach committed by either of the parties absolves the others; and authorises them, if they please, to pronounce the treaty violated and void." A union entered by consent, on equal footing, under compact terms, is a union a state can lawfully leave.

The honest part

We should be straight about the counterargument, because a careful opponent will raise it. Madison, in correspondence, also drew a line between a conditional ratification, which he rejected, and a declaration of reserved rights, which was accepted. He wrote that the Constitution required adoption "in toto, and for ever." Our answer is the distinction Madison himself drew. Virginia, New York, and Rhode Island did not ratify on the condition that they could leave. They ratified fully and stated, as a reserved right of the people, the power to resume their authority later. That is the kind of reservation Congress accepted, and equal footing carries it forward. The reserved right of the people is the live claim, and it is the one that holds.

The bottom line

Texas joined the union on equal footing with the original states, three of which reserved the people's right to resume their delegated powers, with Congress's acceptance. Madison described the union as a compact of consenting sovereigns. Texas's claim does not depend on a single odd clause. It rests on the structure of the founding bargain itself, applied equally to Texas.