I believe that the person who they hire has to be from a Catholic background. Therein lies the hatred of Celtic and Rangers.
It was only Rangers that had a strict "unofficial" rule about religion. Celtic have always signed players from both faiths. Kenny Dalgleish was a Protestant and a boyhood Rangers fan, Jock Stein was also a Protestant.
In fact I remember reading a few years back how in the 50's/ 60's/70's Celtic used Rangers rigidity for their own benefit by actively trying to sign up and coming Protestant talent knowing that Rangers would never sign a Catholic, so they could sign up all the best young players.