The song that Beaver sings is called "The Monkeys Have No Tails in Zamboanga," probably written during the Spanish-American War or the Moro Rebellion. It was popular with US soldiers in the Pacific in World War II.
Eddie gives his full name as Edward Clark Haskell Jr. The "Jr" and middle name contradict information given later in the series.
According to the Patrol Patch on Wally's Scout uniform, Wally is a member of the Beaver Patrol.
In the gymnasium where the boys sign up as Boy Scouts, there's a "To Showers" sign, with a very distinctive arrow, on the wall. The exact same sign is also seen at the country club as Ward speaks to June on the phone.
John Hart and his son Buddy Hart both appeared in this episode but not in the same scene. Buddy played Chester whom Beaver asks if he can play baseball but he tells him to get lost. John Hart is the scout master who recruits Wally, Eddie and Tooey.