If you are living in base housing and there is a shut down you are safe. They will not evict for the shutdown. What will happen is that when the back pay comes through, you will be responsible for bringing your rent up to date.
I'm trying to find the "official" info about this, but that is what I know. You're not in danger of being kicked out, so try not to fret. Just know that you will have to cut a check for the back paid rent.
I've got some information about Atlantic Housing Communities...
"POSSIBLE GOV'T SHUTDOWN STATEMENT
We understand the concern around the possible government shutdown and believe it will be resolved shortly. Once a formal announcement has been made by Congress we will provide further information on how or if it will affect our housing operations. In the meantime, please know that we are working closely with our Department of Defense partner on this and rest assured no one will lose their home due to any delay in pay."