I just talked with the BIA clerk’s office about how it will operate tomorrow if the government shuts down. They said that in-person filing will not be possible, so do not take your notices of appeal or briefs to their office to file them in-person.
The clerk’s office will receive Notices of Appeal and briefs that are sent in the mail to the BIA, but these might not be stamped in. I was told that if the documents are received by the BIA in time, they will be treated as timely filed whether they are date-stamped by the BIA or not. This is, again, a good reason to use a service that confirms delivery (like FedEx or UPS).