"Any of them will do."

"Any" also occurs in positive conditionals: "If you've got any". It's used anywhere that isn't an ordinary positive statement.

I think the "any" in "Any of them will do" -- oh look, and "anywhere that" -- is a quantifier, like "every". It's the difference between "I want to go somewhere warm" and "I want to go anywhere it's warm": one is some indefinite place, the other ranges over all such places.

The dialectal use of "anymore" must be a reinterpretation of some aspect of this -- non-real, or quantified, or something, but I couldn't see what was the bit of grammar or meaning that was actually doing it.