A/C clutches don't fall off. They either fail in the engaged or unengaged (freewheeling) positions.
What is left on the front of the compressor? Did the clutch parts seperate?
Any way it goes, you need to check to see if you can spin the compressor. If you can't, you need to replace it, plus the drier/accumulator and the orifice tube if it has one (that car might use an expansion valve). Then you need to do a serious evacuation (pull vaccuum on the system for a long amount of time) and then make sure a proper amount of oil is put back into the system.
Backyard compressor jobs have a lot of repeat failures. It's pretty much always because of metal and/or moisture in the system (the reason it needs a lengthy evac) or because someone was under the mistaken impression that said evac along with compressor replacement did not remove a substantial amount of oil from the system.