I don't want to get pegged as part of the outlaw group supporting Texas independence, nor do I think that's a very good idea.
But being curious as to what that group based its claim on, they're basically right.
Texas was, without question, an independent nation, and recognized as such. They negotiated a treaty to join the US. The treaty was validly passed in Texas, but it never got the 2/3rds vote required by the US Constitution, which is required for approval of a Treaty, basically because of the controversy over slavery at the time.
So the Senate called it something other than a treaty, and slipped it in under a slim majority vote.
In hindisight, that's ludicrous. It was a treaty, and Texas was never validly admitted as a state.
However, Texas was happy to be admitted, and to be fighting it today is only fun in the abstract and for sport.
But to a stickler for details, Texas is not validly part of the USA.