A green card number or other information regarding your immigration status may be asked for while shopping for insurance. Sometimes that information will need to be physically mailed or fax. Don't worry, this information is completely private and will not be seen by any other agency - it is only to be used to determine eligibility for health insurance through HealthCare.gov.
More information is included in the fact sheets both English and Asian languages.English Fact Sheet
(from Covered California, Asian Americans Advancing Justice, Action for Health Justice)