1992 Corolla Differential Probelm

Our 1992 Toyota Corolla stopped running on the turnpike. Towed it to a repair place that says for $500 they can tell us if it has a problem with the differential. Wondering if it is worth the gamble for a car worth just $4,000. They can't seem to guess what the final repair cost might be. Does anyone out there know?

is the engine works? is the engine run when you crank it? need more information please.

When my differential went out on the highway, the car jerked to a stop. Trying to put it in drive after that caused a grinding noise/clicking and little to no movement when I pushed on the gas. $500 just to diagnose the problem is a rip off. I was charged $650 to fix the problem. Have them simply pull the drain plug on the diff. and see what comes out. That ought to give them a good idea of what happened. If there is no fluid or if there are metal parts, etc. then it is likely the differential. It is an automatic without overdrive, right?
You should be able to find someone else to diagnose/fix it for cheaper even if you have to tow it to another place.
Good luck.

PS-4k for a '92 might be overestimating its value, unless it has less than 40k miles or something

Sounds like either the wheel hub or the CV joint to me.