I can relate my experiences as having been an USFS employee during several govt. shutdowns. Only those people identified as "responsibilities deemed essential" are allowed to continue to work during a government shutdown. The last shutdown under Obama was a bit different, because many things did not shut down. But under previous shutdowns, very very few employees were considered essential. In federal natural resource agencies, this was mainly the key line officers (Park or Forest superintendant and a handful of District Rangers, and Law Enforcement Officers). District Rangers are line officers (usually 4 or 5 per national forest) They are NOT what people normally call a "forest ranger", who are just regular employees in uniform.
So when there is a govt. shutdown, almost all the employees at the park or forest are forced/mandated to not work, and cannot even come in to work and volunteer their time. The park or forest works with a skeleton crew.
That has been my experience through several government shutdowns.