I finally figured out a way to get this done. I tried doing it on a 12 ton Harbor Freight shop press, but there was no way to do it, since the bushings are flanged and must go out towards the outside of the part. Hammering them out was equally hopeless.
Things you will need:
2 jaw puller (I used Powerbuilt Kit No. 43 from Advance Auto Parts)
Some metal tubing that is close to the right size to slip inside the aluminum diff mount's ID
First, remove the diff mount from the diff and subframe. The Mazda manual and the Energy Suspension instructions are good for this part. A fine toothed ratchet will make this easier. I ended up bending the muffler heatshield out of the way temporarily to get a cheater in there to break the four nuts loose. I used a floor jack to control the height of the diff. (I had the entire rear suspension, axles, and exhaust out for other work when I did this)
Now that the diff mount is out, go find a piece of metal tubing that will cleanly slip into the ID of the diff mount, and be able to drive out the metal shell holding the bushing. I got a piece of steel tubing from a welding shop, I forgot the exact size. Just take it with you and try find something close. My tubing's OD was just a bit too big, a mm or so. I used a flap wheel on a grinder, carefully made small flats along it, then smoothed out the flats, rotating the tubing in the vise to gradually reduce it's OD. Keep checking it after doing a full 360 around it, I think I did quadrants of 8 flats each. It took a couple hours, but it will get there eventually. The expensive Diablo flap wheel from Home Depot works a lot better than the Harbor Freight ones too.
Next, stand the diff mount up in a padded vise, and using a hacksaw, saw almost all the way through the bushing flange on two edges, removing about 5-6 mm from each edge. Using the hammer and chisel remove the flange pieces. It should look like this picture. The idea is to create a place for the puller jaw to rest while it does its thing. Do the same thing to the other side. https://imgur.com/gallery/SVVAGre
Now take your tubing, and find some strong metal plate or disc or something for the puller to push on that covers the tubing. Support the diff mount in the padded vise, the bushing you wish to remove should have the flange pointing down towards the floor. Set your tubing (wipe it with some grease or oil to prevent galling) and the plate onto the bushing shell. Assemble the puller, be sure to grease the threads on the screw. Set it up with the jaws resting on the spots created in the previous step, and the screw resting on the plate. It should look like this picture.https://imgur.com/gallery/KO7VMnq
Get cranking, (you're wearing safety glasses right?) and if you aren't wearing safety toe footwear you might want to stand an arm's length away; when it finally comes out the whole shebang is going to hit the floor. Once the first one comes off, clean and re-grease the puller screw and the tubing, and do the other side.
Next clean up the insides of the diff mount with some emery paper or scotch-brite. Use a file or something to remove your tool marks from the chisel. Clean everything again.
Now you can install the bushings like the instructions say. Put it all back together, and enjoy your less squishy zoom-zoom.