There is no connection between the two. This is a question to a labor and/or tax attorney. In essence, the answer depends on the level of supervision which usually determines whether the person is truly an independent contractor or an employee.
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I cannot address the tax law, but can explain the immigration consequences. Whether he can work as a 1099 depends on how he qualified for the green card and how long he has had it. If the green card was based on an employment relationship like a PERM, he is obligated to work for the sponsoring employer as a full time employee. If his immigration or the green card is family based, it is up to him whether he wants to work as an independent contractor or employee. Under either employment status he will have to complete an I-9 and pay his taxes.
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