I think if the congress refuses to fund the ‘nonessential’ parts of the government, it would probably be possible for the chief executive to follow their lead by shuttering those departments and permanently laying off the employees.
What I heard on Fox (I believe) was that the essentials were not being paid and that bothers me. The non-essentials I really don’t care about, leave it closed until the wall is funded, then evaluate their usefulness and get rid the whole bunch.