I've found with Ejecta and Game Center - when you run in the Emulator it will ask to sign in to Game Center but it will fail the first time. When you run the game in future it should then give you a Game Center 'not available' error in the console log, but still run the game OK. Game Center functions will work on a real device but only in Sandbox mode. Make sure you go to Settings on the device and set Game Center to 'Sandbox'.
A black screen indicates something else is wrong however. Can you run a basic project in Ejecta e.g. just displaying a few sprites?
Oh, make sure to use the Ejecta plugin for Game Center - not the native one, as that only works on a PhoneGap/Cordova build.