From Practical English Usage by Michael Swan -
We can use nowhere at the beginning of the sentence for the purpose of emphasis, and then auxiliary verb will precede the subject.
Nowhere had I less expected to see her than in my house.
In your example sentence, Nowhere in the United States but in New Jersey is a negative adverbial. There is the subject and are is the verb. So if it follows the same pattern it will be something like that -
Nowhere in the United States but in New Jersey are there so many people per square mile.
[I don't know if we use there are in the blank, it would be incorrect, but surely the stress factor would be missing. And it's recommended to use the inversion there. So it's better to stick to that.]