At the pool party, Roger introduces himself as "Applebees McFridays": a portmanteau of three large chain restaurants whose trade names are all possessive proper nouns - Applebee's, McDonald's, and TGI Friday's.
Stan seeks the assistance of Dr Olestra Montezuma- whose first name is the same as the popular fat substitute that was initially believed to cause abdominal cramping and loose stools in some people who consumed potato chips fried in the product. His last name, Montezuma, is possibly a reference to Montezuma's Revenge, a colloquial nickname of Taveler's Diarrhea (found on Wikipedia) where residents of more-developed countries travel to less-developed countries and sometimes get sick due to eating/drinking local foods/drinks/untreated water.
Roger's disguise in the opening titles is Tearjerker, from the episode of the same name.
The title of the episode alludes to "An Occurrence at Owl Creek Bridge", a short story by Ambrose Bierce. In the original story, and subsequent screen adaptations and re-tellings, life and a death is not as it seems.