Civilian teachers are federal workers. Whether or not they remain on the job depends on if they are classified as "essential".
Keep in mind that if a Federal worker is furloughed due to the shutdown they MAY NOT work. At all. Including emails etc. It is in fact a Federal offense for them to work without getting paid.
My understanding is that the Military will remain on the job but will not be paid until the impasse is ended. Before you get all excited about missing that Cadet paycheck or ROTC stipend, think of all the military families who live paycheck to paycheck, are deloyed and will not be paid on time. This is unconscionable, IMO.