Normally, this is an Outlook 2013 issue and is also present in Outlook 2010 to a (usually) lesser extent. If there is a policy (Active directory or Exchange) that prohibits the saving of passwords the password won't be saved despite the fact that the option is still selectable. Microsoft has indicated that its the behavior of the interface. When you change connections from the internal network to the external network Outlook sees that as a new connection (even if its to the same Exchange server from the same workstation with the same user ID) and will prompt you again for the password if that policy is set.
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