TodoToday does not directly support automated authentication and login for users that regularly sign-in Teambox using Twitter authentication.
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 for use with the TodoToday for Teambox app. Your Twitter account and ability to login to Teambox.com using Twitter 1-click authentication is unaffected.
TodoToday for Teambox login with Google account
TodoToday for Teambox login with Facebook account