It's REALLY important to link your burrow so you don't lose your progress if you have to delete your app for any reason. Here's how (from the Tunnel Town support pages):
FACEBOOK (for Android and iOS users):
Linking to Facebook is a great way to achieve some additional goals for star bonuses. You will be able to receive star gifts from other Facebook friends who play Tunnel Town and send them gifts as well. You can also share birth announcements on your Facebook wall when you hatch a new baby bunny! You even get stars for sharing bunny birth announcements on Facebook!
To link your burrow to Facebook, follow these steps:
1. In Tunnel Town, click the settings button in the bottom right corner of the screen, it looks like a cog, then click the "my account" icon.
2. Tap the "link" button under the Facebook icon. If you are logged into a Facebook account on your device, a pop-up will appear with your Facebook profile name and profile picture. If this is the Facebook account you'd like to link your burrow to, tap the "yes" button. You should then see a pop-up that says "Social Success", tap "OK". This will link your burrow with your Facebook profile and back up your progress online. If you are not logged into a Facebook profile on your device, log into the Facebook profile you'd like your burrow linked to via the Facebook application on your device, then repeat the steps above to link the account with Facebook.
GAME CENTER (for iOS users only):
To link your burrow to Game Center, follow these steps:
1. In Tunnel Town, click the Settings Button in the bottom right corner of the screen, and then click the "my account" button.
2. If you see an icon with your Game Center name on it, tap "link" and it will link your burrow to that Game Center account. You will receive a pop-up notification that says "social success" if the link to Gamecenter is successful.
3. If you don't see a button with your Game Center name on it, close Tunnel Town and open the Game Center app on your device.
4. Sign in to Game Center, and then open Tunnel Town again. You will have to repeat the first two steps at this point. Your burrow will then be backed up online via Gamecenter once you receive the "social success" pop-up.