You can now use Open Authentication 2.0 in Mailbird. As for now it is available only for Gmail accounts and oAuth is set by default when you add Gmail account to Mailbird. Here's more detail about it.
Just add your name and Gmail address in Menu -> Settings -> Accounts
Click “Continue” on the next window:
Click “Allow” on the next screen:
And that’s it! Your account is now using oAuth for Gmail!
Alternatively, if you prefer using login and password for authentication, you can do it. Just click “Edit server settings” on the “Settings found” window:
Then switch to “Username and password” instead of oAuth:
Then input your password and click “Continue”