Shop is lying.
Literally not possible for the whole exhaust system to clog, and even if it was possible, 90% of the system is open pipes you could clean out with a stick and a brush. It's most likely that the Diesel Particulate Filter is clogged, that will completely plug the exhaust, stopping the engine from running. IDK if a clogged DPF can be cleaned out (I'm sure someone here can tell you if/how), but a new one (Mopar 68417517AB) looks like about $1,800 from Mopar, and it shouldn't take more than two hours for a muffler shop to install. There may be other sensors/misc. parts and programming.
For $21,000 that exhaust better come with a new engine.