Lots of important services and projects may get stalled during a "Government Shutdown" (including E-Verify!)
However, displacement of Americans citizens will continue according to NBC News—Green Card applications are still being processed:
The majority of the Department of Homeland Security's employees are expected to stay on the job, including uniformed agents and officers at the country's borders and ports of entry, members of the Coast Guard, Transportation Security Administration officers, Secret Service personnel and other law enforcement agents and officers. U.S. Citizenship and Immigration Services employees would continue to process green card applications.