OK, sounds like the long screw down the middle of the muffler broke. This is common with OS engines/mufflers, at least in my area. OS has told me that it is caused by the casting expanding more than the screw,when hot, and snapping the screw. They blamed me for overtightening the screw even though I had never touched it and basically said "too bad, buy another muffler from us". The muffler pieces are not available separately except for the screw/nut assy. If you have lost the back piece, you will have to find one on your own or replace the muffler entirely. The disk that fell out IS a baffle and although it usually dishes to the front (I think) I've seen it both ways. If you get a new muffler, go to Home Depot and get a piece of 6-32 All-thread rod and a couple of metal locking nuts to fit. Cut the rod to the length of the muffler and replace the screw with it. Use a nut on each end. You will have to drill the muffler hole open just a bit for the rod to fit. Several of us use this trick and have never had a failure of a muffler modified this way.
The engine doesn't like to run without the muffler because it is tuned to run with the backpressure to help keep the glowplug hot.