I have the same issue as the person who originally posted this question.
I have been playing the Minion Rush game for a long time now. The game requires connection to the Game Centre in order to be able to play with Friends. Suddenly, overnight, could not play it anymore because connection to the Game Centre was lost, and with it I lost the game connection to Facebook Friends. First I thought it was something to do with the game. but trying to connect to the Game Centre from the Game Centre App makes me think it's the problem with the Game Centre. When I click on 'Friends' icon in the bottom bar of the main screen of the Game Centre the message comes up 'unable to load data due to network connectivity issues or errors'.
Outside the Game Centre - no issues at all. I can connect to Facebook, the Internet and WiFi both work fine. I was able to download the recent Minion Rush Game update from the App Store without any problems. I have removed from my Iphone 5S other apps I don't use, deleted lots of photos to free up space. closed the game, closed all apps, restarted, waited, started again. To no avail. The game support people have no clue what else to do.
I note that there are few people on the net searching for an answer to this same question. but not one saying that they have managed to fix this issue and how.
Would your suggested solution (to reset the device by holding the home and the power buttons together) delete all the data and software that I currently have on my phone? I'm hesitant to try it, as I may loose more than just the Game.
PLEASE HELP