This story seems to come from a suggestion made after the Cold War was already over (1992 in the Journal of Humanistic Psychology) but it was never implemented as actual policy.

Let me backtrack a bit to what the military did have: the Personnel Reliability Program (PRP). This was first established during the Cold War for those in the USAF with access to the nuclear arsenal but eventually extended to other areas, like Navy submarines with nuclear weapons.

Psychological "reliability" was required to be authorized in such positions. In a study of 100 cases of submarine personnel referred for evaluation in 1967, here are some of the diagnoses:

Psychotic reaction: Paranoid (3), Acute undifferentiated (1), Chronic undifferentiated (1), Manic-depressive (1), Psychotic (2)

Neurotic reaction: Anxiety (7), Conversion (1), Obsessive-compulsive (1), Neurotic depressive (9).

Personality disorders: Unspecified (12), Psychopathic (8), Passive-agressive (5), Passive-dependent (6), Inadqueate (4), Paranoid (2), Schizoid (1), Cyclothymc (1), Obsessive-compulsive (1)

Not all of the above were disqualifications, as there was room for nuance; only 2 of those with "anxiety" reactions, for instance, were disqualified.

Of the 100 people in the study showing some sort of psychological issue, 48 were returned to service without disqualification. (All 8 of those found to be "sexual deviants" were discharged from service entirely, because this was the US Military circa 1960s.)

Note religion is not considered a factor. So where does the fundamentalist Christian story come from? It's fairly explicitly outlined as a potential disqualifier in a paper by Stephen Kierulff. From the abstract, which I'll quote in its entirety:

Millions of Americans, primarily premillennialist fundamentalist Christians, believe that God has foreordained a global nuclear war as the precursor to the Second Coming of Christ. Apocalyptic religious beliefs would be of less consequence were it not for the fact that after being given computer warning of an apparent nuclear attack, U.S. personnel in the midst of the electronic loop have just a few minutes to decide whether or not to launch missiles in retaliation. Early warning computer systems have malfunctioned in the past, and religious convictions about the inevitability of a nuclear war could incline a person to make the disastrous error of believing an erroneous computer warning to be correct. In the U.S. military, access to nuclear devices and authorization to participate in their "delivery" is controlled by the Personnel Reliability Program (PRP). For the sake of global safety, American psychologists and psychiatrists, under the auspices of the PRP, should be engaged in screening out personnel who are convinced that a nuclear attack against Russia would accord with God's will.

A reminder, though: this paper was published in 1992, and it is a suggestion -- it states that psychologists "should" be screening in this scenario.

This suggestion was never implemented in any version. You can read the DoD's current policy here.

...

Kierulff, S. (1992). Armageddon Theology and the Risk of Global War: The Limits of Religious Tolerance in the Nuclear Age. Journal of Humanistic Psychology, 32(4), 92–107. https://doi.org/10.1177/0022167892324006

Satloff, A. (4 Oct 1967). Psychiatry and the Nuclear Submarine. American Journal of Psychiatry, 124.