Generally, you need to include just in this construction. For instance, in two of your examples, removing just changes the meaning:

They won't let any person in
We don’t publish any letters

Without just, these statements indicate that nothing whatsoever is permitted; no people are let in and no letters are published. If you put the surrounding context back in, the meaning becomes evident, but then there's a semantic mismatch and confusing language.

Ellipting just works better in the pepper example, because the construction is [X], and not any [X] but [specific X]. The context before and after (mostly after; but does the heavy lifting here) not any makes it obvious there's an exception to the rule of not any pepper, so it's not nonsensical like the other two. However, it still reads better with the intensifier, because just helps emphasize how special and important the Indian pepper is, which is the purpose of this phrasing.

An equivalent phrase for just any is any old. This is idiomatic, with old meaning, roughly, generic, and not aged. This is informal, and as with just any, removing old changes it into a blanket statement.