I have just had the same problem and after much struggling have managed to fix it. The key is you have to get an app specific password from your google account (your standard password will not work). Hopefully these steps help
1) Click onto your google account (top right hand corner of your browser if you are logged in, round circle)
2) Select My Account
3) Select Signing Into Google in the Sign In and Security box
4) Click App Passwords
5) You may need to sign in again for security (standard password)
6) Select YOUTUBE in the drop down box at the bottom for select app
7) Select the device you wish to logon with the other dropdown box
8) Click Generate - to generate a new additional password
9) Use this new password with your existing username in FCPX and it will login to the YouTube Account.
Good luck!