Its likely a fairly complicated answer that depends on how you are paid and state laws concerning unemployment (they vary). Many interns are paid as contractors (e.g. 1099-MISC vs. W-2) which generally means you aren't eligible. Of course, whether or not that is appropriate is a separate issue but a couple months pay is probably less than the legal fees necessary to get that sorted out....
Does this apply to international students as well? I heard it gets easier to apply for a permanent residency after getting your PhD but I imagine that would be after your internship even if you defend your thesis before going on internship?