I also understand that the Ohio-based company Elmer's did release some innovative synthetic glues in the 1940's shortly before this song was written, but from a brief search I see no connection with this period of Elmer's history to the state of New Jersey.
Also, are there other examples of commonly-held state stereotypes that have evolved or were lost across the decades?
The glue reference is to horses. In the US, horses used to be slaughtered and their collagen was used to make glue. Injured racehorses were often used for this practice because they have huge muscles (where collagen comes from) and were no longer much use to humans. :( I can’t find when this practice was largely stopped in the US (most companies weren’t like “hey guess what, we killed horses to make this”) but there have been no legal US horse slaughterhouses in existence since 2007. . However, horses can be shipped overseas where they still may be used for animal-based adhesives.
After World War 2, NJ was known as a hotspot for horse racing (and still is, to some extent). In the early 1940s its Golden Triangle of racetracks opened. Horse racing was more widely followed than it is now because it was one of the only forms of legalized gambling.
Did NJ have more slaughterhouses than elsewhere or provide the US with a prolific amount of glue? Although not necessarily untrue, I can’t find anything in particular to back that up. But since it’s just supposed to be a funny song, my guess is that the song’s writer just made the leap from lots of horses to glue. It was a common saying that when a horse had had already hit its peak or was no longer useful it was “off to the glue factory,” a phrase you still occasionally hear today.