In Lighthouse, Jack's mom says he got his appendix when he was about 7. I am not sure when Jack was born, but if he was born in 1970, he would have been 7 in 1977. This is, of course the same year that the isalnd was (assumedly) sunk. There is probably some significance in that. It kind of begs the question, what about the island being sunk caused Jack to have his appendix removed?
One thing is for sure, Jack was "destined" to have his appendix taken out.