To be fair to them, I think they may be half right. They talk about "Lack or loss of spiritual faith" as being indicators for suicide.
"Lack of", I agree with everyone here, is not an indicator for the reasons given.
But "Loss of" - particularly sudden or rapid - I could see as being a valid indicator. Someone who has held a god belief for a long time, and who sees the things a serving soldier would see, who rapidly questions and loses his faith in an environment where they have easy access to weapons....yes, I WOULD put that on my list of things to watch for.
When Holy Joe the PFC hurls his Bible away and starts to ask "what is the point?", then yes - that IS something his CO needs to take notice of.