Can a loose wire on your lighter cause your radio and car to shut down

I never heard of the engine shutting down but the radio's memory circuit is always tied to something that is always hot such as the interior lights, horn, or lighter, (if your lighter works without having to turn on the ignition switch). Many lighter sockets have special contacts that match the lighter element but other power plugs have mismatched contacts that, in some instances, can short the socket and blow a fuse. You just have to be careful how you insert the plug or use the dedicated power receptacle a lot of cars have now.

Check the fuses in the car and under the hood.