Ok, I've never seen what you're talking about but I've replaced literally hundreds of suspension and other bushings so...

I'm assuming that this is a typical rubber bushing with a steel sleeve in the middle for the bolt to go through and a steel outer ring that is press fit into the housing.

Sometimes when a bushing fails the rubber will rip and the middle will fall out. If not, try to cut it out.

If you don't have a proper puller, use a hacksaw blade and carefully cut a slot into the outer steel part of the old bushing from the inside out. Try not to cut into the diff, but even if you do, the bushing is supported 360 degrees so a small nick won't hurt. Once you have cut through the metal it should come out easily. It may even just fall out.

Don't forget to grease the outside of the steel ring and the inside of the bore before you install the bushing.