I recommend referring to the following forum post on how to achieve this: Re: Secure Zone - show, hide, or disable content when a user is recognized as logged in.
To summarize, you would need to use the module {module_isloggedin} which will render a 0 or a 1 based on whether the customer is logged in or not. With a combination of CSS/JS, you can specify a different menu to display based on whether they're logged in or not.