Let us think this one through, if we may. The "Former Occupant of the Oval Office, 1993-2001" could snap his fingers, and ready toadies in the Irish Parliament would have his application for Irish citizenship in the Prime Minister's desk for approval the next morning. Once established as an Irish national, the "Former Occupant of the Oval Office, 1993-2001" would be free of that nagging obstruction to his goal of becoming Secretary-General of the UN, a limitation that is placed on any national from the permanent members of the Security Council (citizens from countries that are revolving members of the Security Council seem not to be subject to the same restriction). Plus, there are a whole new bunch of babes at close proximity, and none of those nettlesome bourgeois rules about sexual harassment at the UN. Why, with the opportunities concerning Iran and North Korea concerning ways around international sanctions, it could be a highly lucrative position as well, paying well beyond the typical UN salary.
For the "Former Occupant of the Oval Office, 1993-2001" I just don't see a downside. For the rest of the world, not so good.
There is no strict rule about citizens from permanent members of the council. It's just a tradition. But U.S. has veto power over the nomination.