Unfortunately, the way things sit right now, I doubt you'll see much help from the dev's, as it has become pretty much a finger pointing blame game which is really too bad because this game had the potential to be great. However, with Telltale (the publishers of the console version) going out of business, I doubt you will see any response. The Fun Pimps, (who wrote the game and are responsible for the PC version), are out of the loop on this regardless of how they attempt to portray their involvement. The console version operates completely different from the PC version and I wouldn't be surprised if no one at TFP had a clue about how the console version operates.
For my part, all I can suggest are the following:
1-Make sure you reset/clear the cache on your xbox, (shut down completely and unplug for ten minutes-some say five. I don't hold much hope for this one but it doesn't hurt to try.)
2-With 7 days selected but not started try pressing the menu button (the three line button on your controller) and see if it brings up the game menu. If it does, try deleting all local saves and restart the xbox.
3-I am assuming you have an active xbox live account. So I would suggest you disconnect from the internet and try loading for a local game.
4-Contact who you purchased it from and ask for a refund. Unfortunately, this is the only likely solution and depending on who you bought it from, and their policy, you might get your money back.
As adept as The Fun Pimps are at creating a game for PC, I don't believe they have any clue how to deal with the mess that has been created here. It was painfully obvious early on that Telltale had no clue, and Iron Galaxy, who actually ported the game, appear even less concerned about the game than TFP.
Best of luck to you.