once employed, if you revealed in conversation your true beliefs, and
they ran counter to what you wrote down, could this give the employer
just cause in termination?
Texas is by default an at-will employment. Thus, absent a contract establishing termination for just cause, the employee may be terminated for anything (except when that contravenes public policy) or even for no reason at all. But you specifically ask whether the employee's eventual disclosure constitutes just cause.
It depends on whether the employment contract specifies that "the employee would be employed for so long as he satisfactorily performed his duties". Hardison v. A.H. Belo Corp., 247 S.W.2d 167 (1952). In that event, the sole discovery that employee lied about that at the interview is not evidence that the employer was dissatisfied with the employee's work performance. In a context of termination for just cause, Porter v. United Models, Inc., 315 S.W.2d 340, 344 (2008) states that
where performance is to be the satisfaction of one of the parties, his
dissatisfaction must be founded on facts such as would induce action
on the part of a reasonable man. He may not act arbitrarily or without
reason in the matter, and the law will say that he is satisfied with
that with which he ought to be satisfied.
Ultimately, an employer seeks to be satisfied with the employee's actual work performance, rather than with a screening process aimed at predicting the person's work performance on the basis of the employee's personal values.