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The default presumption is that employment in the state of Texas is "at will" absent an express agreement to the contrary. At will employment can be terminated for any reason not amounting to discrimination on the basis of a legally protected trait (race, religion, national origin, etc.) or retaliation for engaging in certain forms of legally protected conduct (filing a wage claim, taking FMLA leave, etc.). It doesn't matter whether the basis for termination is fair, reasonable or even true.
So, there is nothing inherently unlawful about laying off an employee and reassigning that employee's job duties to another employee who happens to be working on an H1B visa. What would be illegal, however, is if the employer's MOTIVATION for such decision was the employee's national origin. This is because national origin is a "legally protected trait," and it is illegal for employers to base employment decisions on legally protected traits.
The challenging thing is in proving motivation. The burden of proof would fall on your shoulders as a plaintiff, and you would need to be able to establish by a preponderance of the evidence that "but for" this other employee's national origin or your own, that you would have been retained. That can be very hard to prove, and it may be that it's impossible to prove if this wasn't your employer's motivation to begin with. Nonetheless, this would be your sole avenue of legal recourse under the circumstances, typically speaking.
If you believe you can prove that you were let go due to national origin, your recourse would be to file a complaint with the TWC Civil Rights Division. The Civil Rights Division will conduct an investigation and attempt to mediate a resolution with your employer. That not forthcoming, they will either file a lawsuit on your behalf or issue you a "right to sue" letter which will authorize you to file your own lawsuit in civil court with the assistance of a local attorney. See here for information on the complaint process with the Civil Rights Division.
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