Updated information on how to prevent an ESTA or Visa overstay in the United States:
Travelers who are in the U.S. on the Visa Waiver Program should call USCIS Customer Service (1-800-375-5283) or use the USCIS Contact Center just before, or on, the expiration date of their ESTA if they cannot find a flight to leave. The Customer Service Desk will help them schedule an appointment at the nearest USCIS Field Office, usually for that same day. These Field Offices are closed for regular operations but are open to grant VWP extensions for a maximum of 30 days for Satisfactory Departure. If after the 30 days they still cannot find flights, they can go through the same process for an additional extension.
In exigent circumstances, travelers can go to the following website which provides a phone number (202-325-5120) for the CBP Traveler Communications Center. There, staff can provide case-by-case instructions. https://www.cbp.gov/travel/international-visitors/i-94/traveler-compliance
Non-ESTA cases, i.e., those who are in the United States on a valid B1 or B2 visa and are concerned they will be unable to leave the U.S. because of the current situation, should file an extension of stay request by submitting form I-539 by mail, along with form I-912 to request a fee waiver, an explanation why they are unable to leave, and any other relevant documentation. Instructions on where to file can be found here: https://www.uscis.gov/i-539 addresses. Those who are already very close to the expiration date can also call USCIS Customer Service for further instructions.
Callers should have their I-94 Departure record number available when calling, that is usually stapled in their passport (which was provided when entering the U.S.)
You should be able to talk to a USCIS representative by calling +1-800-375-5283 and pressing the following options:
Press 1 for “English”
Press 2 for “Otherwise”
Press 7 for “All other information”
Press 2 for “Naturalization or Visitors”
Press 2 for “Non-immigrant aliens, foreign students, and visitors to the US”
Press 2 for “Extend status”
Press 6 to “Talk to a representative about this topic”