Answer: The people who had the right to claim Texas is the mexicans. The reason for this is the Mexican were the first ones to INVITE the settlers to Texas in the first place.
Millions struggled and some even lost their jobs. Mexicans wanted to give americans part of their land, multiple times they tried making it apart of Mexico. People were paid to bring settlers to their land.
For instance, Empersarios were the ones used to bring settlers to Mexico. And Stephen F Austin had became the one to form a colony and call it, "Old Three Hundred." The effect that was given made a big impact for Mexico.
Mexicans had been the ones that won the Independence of Spain in 1821. Therefore, Mexicans were the ones that had the right to claim Texas.
Hope this helped.