"Never say die?"

The expression "never say die" has been around since at least the early 1800s, but I think OP is attempting to over-analyse the grammar.

It just means don't use the word "die" (as in "We're all going to die!") because that would imply you've given up hope, which might become a self-fulfilling prophecy. It's much the same as "Where there's life there's hope"

As @Edwin comments, there's also Never say never. Add to that a couple of thousand written instances of Never say can't (and a couple of dozen Never say hopeless, for example) and I think one could reasonably say this is a "productive" construction even today.

According to Concise Oxford English Dictionary, one says “never say die … to encourage someone not to give up hope”.

It functions literally as a reply to a statement such as “we are all going to die”, expressing hopelessness in the face of a deadly situation.

It is also used idiomatically as a reply to similar statements of hopelessness not containing the word die, or when the speaker anticipates such a statement and wants to forestall it, or when the speaker is struggling with feelings of hopelessness and wants to banish them.

The earliest published example I found is from “The-Man-of-War’s-Man, Chapter XVI” (evidently part of a novel serialization) printed in Blackwood’s Magazine, Volume 18 (1825):

Cheer up then, and never say die, for the devil a morsel of good it will do.

To uncover more examples, you can use the Google Ngram Viewer: search for [ never say die,Never say die ]; click through the various date ranges beneath the chart to view the source texts using Google Book Search.