So this may be an oversight but the National Guard falls under Department Of Defense. I am also an Active Guardsman and Department of Defense has a budget and we are being paid. The only Military Branch affected is the Coast Guard who falls under Department of Homeland Security.
I know others have said this is a business, but part of running a business is keeping turnover rates down. Evicting will cost you more than late rent. You should still charge late rent fees but if the tenant is in the Coast Guard I would just explain in writing that due to the government shut down rent will still be late but no eviction will be started until ten days after an agreement is reached. It should take about ten days for pay to be submitted.
Side note for all of the hard core business minded individuals. Coast Guard men and woman are under Contract and cannot just quit and find another job. They can be charged and jailed for this. Should they have money saved? Sure, but 10% of their wages over a year will not cover most rents. And the tide can be turned on you also saying shouldn't you have enough in reserve to cover your vacancy?
Just my two cents!