Yes, the Department of State requires all J-1 scholars and their J-2 dependents to have health insurance in order to stay in the U.S.
The Exchange Program Period that is marked on your Form DS-2019 Certificate of Eligibility determines the length of your legal stay within the U.S.
As J-1 exchange visitor, you are permitted to remain in the country up to 30 days past the end date of your Form DS-2019 . This time is to allow you time for tourism and packing. You do not have work authorization during this time. If you require additional time to complete your project, request both DS-2019 extension, if eligible, before your grace period begins.