So with the latest update, Jetpack Joyride will not get past the loading screen. I am afraid to uninstall and re-install the app because there is the possibility of losing all of my progress and purchases. Does anyone know if I will indeed lose everything or if it will save like some apps do?
Yes and no.
If you delete it, you will lose all your game data, but things you achieved with Game Centre (eg a new hiscore or an achievement) will be kept on your Game Centre account.
An option will appear when you delete the app to keep the app on Game Centre. Enabling this will keep your achievements and such.
I believe there's no way to keep game data: That will get erased.
As long as your save game is not what causing the crash, you should be able to backup your save to a PC/Mac, delete and reinstall the game, then copy back your backed-up save.
I've done this method with an PC/OSX tool called PhoneDisk, now discontinued and repackged as iExplorer. This is the guide I've used, which I've linked in an answer to a somewhat similar question of mine.