One of the staff asks Jack to look at an x-ray of a man's L4 vertebrae. Previously on the show Benjamin Linus had a tumor on his L4 vertebrae as well.
The book which Jack is reading to a sleeping Aaron as a bedtime story in one of the flash-forwards is 'Alice's Adventures in Wonderland' by Lewis Carroll.
The medical maneuver that Juliet performs on Jack to diagnose is called Murphy's Sign. It is not typically be associated with appendicitis but is usually associated with cholecystitis (gall bladder inflammation).
Jack wanting to use novocaine and mirrors to stay conscious during the surgery is based on a real event. During an expedition of the Antarctic in 1961, Soviet surgeon Leonid Rogozov fell I'll with appendicitis. Since he was the only doctor on the trip, he performed the appendectomy on himself using novocaine and mirrors held my other members of the expedition.