It's never really gone into detail, but from what we see and hear of Egg, it can be assumed that he is older and more powerful than he looks. It is also very likely that Egg Shen had fought Lo Pan sometime in the past, evidenced by Lo Pan's line during their magic duel, "You never could beat me, Egg Shen." We do know that Egg Shen is knowledgeable about Chinese history, Chinatown, and Chinese black magic. He's "a very rich man" who owns several large buildings in Chinatown. Egg also runs a San Francisco tour bus service called "Egg Foo Young Tours" where he acts as a bus driver and tourguide. Perhaps Egg is an ancient sorcerer like Lo Pan, only he aged better. Maybe he's just had a chance to face Lo Pan when he was younger. It would take a movie prequel to explain this.
James Dalton, or just "Dalton" to most, is a legendary "cooler" (that's bouncer for those who don't know
Road House). So legendary that bar owners and troublemakes across the country know his name. Dalton is a trained martial artist with the ability to rip out a man's throat with his bare hands. Dalton is a good man at heat, who will always try to end a confronation without fighting, and if he must fight, will do everything to make sure the fight is away from others ("always take the fight outside"). "Pain don't hurt" is perhaps his greatest philosophy.