What is the term for, "Don't make it a thing if it isn't a thing"
Solution 1:
Procatalepsis (or sometimes "prokatalepsis") is the denial of something before it even comes up; it's applicable to any of those "I know what you're going to say..." denials.
But for subtle accusations by others, you're probably looking at apophasis.
These don't cover the specific case of denying the opposite by negating a word though, so what you're describing may be a variety of litotes.