TodoToday does not directly support automated authentication and login for users that regularly sign-in Teambox using a Facebook account.
The Teambox API used by the TodoToday for Teambox application does not support authentication via 3rd party services like Google, Facebook, Twitter or Open ID
To enable the ability to login to the TodoToday for Teambox application you need to use a Teambox specific password for your Teambox account.
If you have not already setup a Teambox password go to https://teambox.com/forgot to request a password reset. This will create a new password that can be used to login to the TodoToday for Teambox app. Your Facebook account and ability to login to Teambox.com using Facebook 1-click authentication is unaffected.
TodoToday for Teambox login with Google account
TodoToday for Teambox login with Twitter account