Shearer has joined O'Neill's critics but the Irishman has the full support of owner Ellis Short and the board.
O'Neill said: "I don't want to sound boastful, but not only am I the best man for the job but the only one.
"I once really doubted myself before I sat my 11-plus - honestly. I didn't know whether I was going to pass it or not, but then I was only seven at the time. And I did pass it.
"Criticism is part of the game. I think it's an easy thing to point to, so what do you do? Put someone else in charge, give him 12-13 games and he can't win any matches? We will steer it through, get calmer waters ahead and then we will see what sort of job I have done.
"We have to win some games. I have said it before. I came in last season and we won some games, some of which I probably didn't have any right to, and some things may level themselves out."
O'Neill, meanwhile, laughed off the assessment of Shearer, a former BBC colleague, who has said Sunderland were a team who looked doomed to stay in the relegation zone.
O'Neill said: "It really doesn't worry me what Alan says. You lose games and you get fired at. He is a pundit. I have been there myself during European evenings and given a trivial amount back. I usually try to give someone criticism that I will never see again in my life - such as somebody playing for Inner Mongolia."