Do rear quarter extensions require a gasket?

Yes, though it's not really a gasket per se, it's more like a mounting cushion. There's a lip on the inside of the extension that a rubber piece fits onto. Obviously, the mounting studs have a gasket to keep the water out of the trunk. I think the parts suppliers carry the "gasket" you're looking for.

If you don't use the gasket, it will eventually work loose and rattle.

Yes, but is more like a strip gasket.