Federal and Texas Law:
Texas' laws with respect to overtime pay adhere strictly to the federal Fair Labor Standards Act (FLSA). Employees who work more than 40 hours per week are entitled to either 1.5 hours of comp time per extra hour work or 1.5 hours worth of pay per extra hour worked. According to this source, though, the comp provisions are for government workers and the over time pay provisions are for non-government employees. In fact,
If you don’t work for a governmental agency and you get “comp time” for overtime hours, your employer may be violating the Texas’ overtime law. For example, if you work 45 hours during the workweek, and your employer tell you to take 5 hours off the next week to make up for the extra hours worked, this is a violation of overtime law. You must be paid for excess hours worked.
"Non-exempt" employees are those who are exempt from the provisions of FLSA. These are typically hourly wage workers. They must be paid minimum wage and overtime pay for any work over 40 in a week. A nonexempt employee whose extra hours are not properly accounted for and reflected in his paycheck, he may find it in his interest to file a wage violation complaint against the employer with the Department of Labor's Wage and Hour Division (WHD), which is explained in further detail here.
Exempt employees don't receive FLSA protections and are not entitled to overtime pay. Employees fall under this category in a couple ways. The first way is by action of law and you can see a list of categories of employees exempt from the law here and here.
The following is possibly the most relevant part for the individual in question, but please note, there are other determinations to consider above and beyond the below-quoted text:
Section 13(a)(1) and Section 13(a)(17) of the FLSA provide an exemption from both minimum wage and overtime pay for computer systems analysts, computer programmers, software engineers, and other similarly skilled workers in the computer field who meet certain tests regarding their job duties and who are paid at least $455* per week on a salary basis or paid on an hourly basis, at a rate not less than $27.63 an hour.
Read the Wage and Hour Division's Fact Sheet on the Exemption for Employees in Computer-Related Occupations Under the Fair Labor Standards Act here for more information to see whether this individual falls under this exemption. If you're still unsure, call WHD at 1-866-4USWAGE. Please note (this info is at the bottom of the fact sheet page) that these provisions may be subject to change, as DOL has been working through the rulemaking process with the aim of adjusting these provisions, although I do not have information on what changes are contemplated.
This Texas employment litigation law firm says relatively clearly that, although at-will employees do not have much power with respect to their work schedules, "Texas law places no restrictions on the hours that an employee can work, but employers must pay employees overtime for hours worked in excess of forty hours per week."
Also, worth noting, from this Texas law firm, because the firing does sound rather arbitrary (although I declined to address the age discrimination issue at this point based on a lack of enough info) and because of the possibility the company is already out of compliance with the law:
Even when it appears that these strict requirements are met, technical legal reasons may prevent an employer from denying overtime pay to a salaried employee. This is because employers must dot every "i" and cross every "t" or risk owing thousands of dollars in overtime pay.
Do You Have A Work Contract?:
If yes, read it! If no, maybe don't be so sure. According to this 2016 article about a case out of Pennsylvania, even an email exchanged between an employer and employee, assuming it contains enough specific material, may be viewed by a court as an employment contract.
One More Thing:
Maybe. As most other things do, this may vary by state and may or may not be applied differently in the context of an employer-employee relationship (or may not apply at all if a separate work contract covers it), but someone in the employee's position in the question may be able to seek and obtain relief through the concept of unjust enrichment. This is often used as a backup when one is operating without an applicable contract. Generally speaking, if someone provides someone a benefit and the second person, upon receiving and enjoying the benefit, does not compensate the first person, one could say the second person was "unjustly enriched." If it can be determined that the Employer in the question was unjustly enriched by the work of the employee, the employee could perhaps be able to be paid restitution. More details on this concept at the above link.