I can’t identify the diner, but the names are familiar: Bill Griffith finds -wick names intrinsically funny, and he’s not alone in this.
Sedgewick in an earlier Zippy here, where (talking about names) I said that “Sedgewick has a number of possible sources, and might have been chosen just for its sound.”
Then there’s Fenwick, notably here:
The Duchy of Grand Fenwick is a tiny fictional country created by Leonard Wibberley in a series of comedic novels beginning with The Mouse That Roared (1955), which was later made into a film. (link)
Why -wick names strike many people as silly is an interesting question in itself, and I don’t have a quick answer. (“Just because it is” is not an accetpable answer.)