Let me begin by stating that I practice employment and labor law, not immigration law. As an employment lawyer I regularly handle matters coming before the Connecticut Department of Labor's - Employment Security Division. If you could provide me with copies of the documents you received concerning your U.C. benefits, I would then be able to properly advise you with respect to only the U.C. matter. Good luck and best regards, Rob Fortgang - Employment Law Attorneys / 860-658-1055 / 800-932-6457 / email@example.com / We can Help.
If you need to show to Unemployment that you are presently authorized to work and have a pending I-360, you may file an I-485 application with an I-765 application to receive an Employment Authorization Document. Please feel free to contact me if you need help.