We all know that Gödel showed that there, in a formal system, are true statements that are non-provable (undecidable). In ZFC, there's Kaplansky's Conjecture, the Whitehead problem, etc.

We can also agree that we're sure to find more non-provable statements in ZFC. What I'm curious about is the following:

What is the defining characteristic of a non-provable statement in ZFC? Are they all "strong" in some sense? Is it a necessary condition that they are strong, then? What future theorems might turn out to be non-provable? Is it, through the characteristics of the theorems known to be non-provable possible to make a fairly accurate guess if a theorem will turn out to be non-provable in ZFC?