I've started a small business and am going to be hiring people for the first time, and am not sure what I can legally ask. Can I ask them why / how they left their previous job? Why they're not eligible for rehire?
Yes, both of those questions are legal in New York State.
Steer clear of questions pertaining to protected class - Age, Race, Gender, Disability,Religion,National Origin, Sexual Preference, Intent to become pregnant or familial situation etc.
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Class Action Attorney
Neither question is legally objectionable.
But from a practical perspective, you might consider hiring through a temporary agency. While it will cost you the commission, you don't have to worry about any of these things. Plus you don't have any commitment, so if the employee does not work out, you can easily ask for a replacement - they usually have people standing in the wings.
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Employment / Labor Attorney
I recommend speaking with a management side labor and employment attorney before hiring employees to make sure you are doing everything right from the beginning instead of cleaning up a mess later.