I'm not an expert or author but have they tried logging in using a computer and a client where you CAN unblock yourself?
Or have someone else who they trust log in with their account?
Just a thought.

"Block" is a client-side function, so the only way to fix it is to do it on that client.  There might be games that the user could play with logging on as someone else from their phone, but I don't know enough about it to be sure.

Of course, if they switched to a different client then the problem would completely disappear unless they also wanted to use the old client.

The author came back with instructions to fix this - they will also work on the app so it wont allow the user to do this in the future 

0) Quit BG Online

1) Locate the Terminal app and open it. You can open it e.g. via Spotlight or Launchpad

2) Paste the following line into the Terminal window and hit return:

defaults read com.multimedia-go.BackgammonOnline bannedUsers

3) This should show you the list of banned players. If you are the only one on the list or you don't care about resetting the list you can just enter the following line and hit return:

defaults delete com.multimedia-go.BackgammonOnline bannedUsers

If you do want to retain the other players in the list, enter the following line into the Terminal and hit return:

defaults write com.multimedia-go.BackgammonOnline bannedUsers -array player1, player2, …

where player1, player2, … are the names of the players you want to be blocked.

4) Quit Terminal.

5) Open BG Online. The list should be updated and your user name visible again.