These aren't legal issues.
You have an offer. It is contingent on completing the application and employment process. You can refuse to provide any info you choose> If you refuse any requested info, the prospective employer may accede to your position or the employer may go away and hire someone else more forthcoming. That's your risk. But none of this is mandated by law. Do what you will and accept the consequences.
Most applicants in this economy would be more than happy to provide the info requested, but YMMV.
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You should repost under contracts and labor law. There is nothing pertaining to education law in NYS in your query.
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Refusal may end in retraction of the offer.
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