Now that this situation has happened to me it lead to some good brainstorming which turned out a solution that actually worked. I limped my vehicle out and was able to do this not on the trail. Knowing what I know now I could do it in the trail if I had the right tools.
Anyhow to the fix. You need a 13/16" Milwaukee hole hawg bit with the spring loaded center punch or similar tool, a drill motor of some sort and the tools required to remove and disassemble a differential.
Start by removing the wheels, brake calipers, and diff cover. Next remove the long part of the broken shaft. Remove the pinion shaft from the differential and push the good axle shaft in towards the diff to access the c clip and remove it allowing you to pull the good shaft out.
Now remove the main bearing caps from the diff and begin to pry out diff keeping the shims for each side together and accounted for. It takes quite a bit of force to pull a diff out I usually use the handle for a 1/2" drive ratchet if a pry bar isn't handy. Once out disassemble the diff. In my case it was a Detroit locker and removing the diff was necessary with a lock right or something you MAY not have to.
When disassembling a Detroit be careful there are 2 large springs that can send the case flying at you face. Take it apart piece by piece until you get to the side gear with the broken piece in it. In my case it was the driver side which requires the most disassembly. With the side gear out clamp it to something solid or in a vice. Use the hole in the center of the shaft to center the bit and start to drill. Drill down until you hit the c clip and the end of the shaft that retains the c clip should come off allowing the shaft to come out.
