The right to organize is a fundamental right of all Americans - it cannot be made "illegal" by any state. That being said, Texas is a 'right to work' state which means that there cannot be any 'closed shops' in which all employees must be union members. The previous poster is correct. There is one (for profit) Houston facility in which nurses recently voted for union representation.
Before you write us off, you should compare work environments, benefits, etc. I believe you will find them comparable to what you are accustomed to.
Organized labor has never been very popular in Texas. Our culture emphasizes rugged individualism and independence than group activity - even for "worthy" causes. Example - our unfailing and absolute dedication to "one person - one vehicle" rather than mass transit even when we know that it doesn't make sense
(yes, I am a Houston native).