Overheating? Throwing some thoughts out here...
Are *both* of the zero's using heatsinks? Especially with a multimedia streaming application? Is the zero actually the best hardware for this project, rather than say a 3x model?
Is one encased and the other one free-standing?
General:
Is the dongle you use to connect to the micro-usb ports for ethernet (assuming ethernet) the same? Could one be defective or loose? Or could one just simply be a corner-case that tips over with a streaming application?
Are there any configuration differences in your config.txt file?
Are you following the advice of the piCorePlayer project that since it uses TinyCore running in ram, it is totally ok to just shutdown by just pulling the plug? On paper, you can. In the real world, possibly not the best idea.