It is your job duties and not how you are paid that determines whether you are an exempt employee. Not all salaried employees are exempt: not all exempt employees are salaried.
There is no law anywhere in the US, where private employers are concerned, that requires that you ever get a holiday or a paid day off. In many states, including Texas, you could legally be required to work 365 days a year. And while I do not disagree that it is shortsighted, that does not change the law.
I do, however, see a couple of potential flaws in the employer's policies. Before I go any further with them, could you give me an idea of your job duties?

