Generally they also need a search warrant, but it depends on all of the facts and what they were told. If they had "probable cause" to believe he did live there it changes things. The law in New Jersey may have stricter requirements than the US Constitution.
If he has been arrested, he needs a criminal defense lawyer, now. If you are upset about the violation of your privacy you should contact a personal injury lawyer.
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It depends on many facts. They can arrest him and look for him at the home. But they cannot search for him based on the information in your post.
They need a warrant. They will probably say you allowed them in.
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