If I prevent the use of a characters attached item, in this case a fragment, and that character KO's a character or hits with doubles, will the resource still click? Also, do they only lose the effect that the assigned item grants, or do they also lose all effects the resource grants?

Equipping an item isn't the same as assigning it. I'm prone to say no

PF fragments aren't equipped, so I agree with RJ.

You can't prevent the use of assigned or attached game elements with Joker's power, so the question is kind of irrelevant. He prevents the use of equipped items. The Phoenix Force, and its Shards, are not equipped items.

Bah, I can't believe I overlooked that wording, thanks for pointing that out.