The Spectral Family is puzzled by the existence of Mount Rushmore. Since they last vacationed there in about 1890, this is true. The monument wasn't completed until the 1930s.
The ghost family as Mount Rushmore sing a song to the tune of Hit the Road Jack in an attempt to get the Ghostbusters to leave.
Egon, Ray, and Janine do not appear in this episode. They are not even mentioned.
On the way, Peter, Winston, and Slimer sing the "Itsy Bitsy Spider".
After the fishing incident, Peter imitates actor John Wayne.