My unit just told me that I have to turn my personably-owned firearms. Do I have to? Does it make a difference if I live off-base?

Generally, yes. The orders of your command will be presumed to be lawful unless evidence indicates to the contrary. Your command has an overarching duty to maintain the safety of the member and others. This duty does not change even if you live off the installation. As with most issues in the law, there is nuance and grey areas. For more information, please read this blogpost.