Not really. Protecting by law ones right to withhold information that would incriminate them is one thing. But the incarcerated guy doesn't seem to claim there's anything incriminating him in these files. He simply stated he doesn't want to share it with the cops.

My understanding is, it's not like the FBI got a warrant for this etc, and instead started flinging shit at him - which is clearly bad. But for this narrow argument, that's besides the point, IMHO (IANAL). Because in the first place, the guy simply didn't want to share this information with law enforcement. There is no claim that it incriminates him.

How that stacks up against actual US case law, I have no idea of course, but I don't see how it follows from the right to not self-incriminate.