Alright... here's kind of the pain in the butt. Is that menu just auto generated from your pages?
Or do you use the custom menu interface?
There may be a function for omitting pages from the menu if someone is not logged in, I'm not sure.
On my site when I wanted different links for logged in and not logged in, I registered another nav_menu in functions.php
Then in header.php I used the is_user_logged_in conditional to display 1 of the 2 menus I had.
In the menu interface I made a menu appropriate for each
As I'm sure you know, just hiding the menu item, doesn't keep anyone from accessing the page. If it's important to you to keep it hidden, there are membership plugins to keep people from accessing the page who aren't logged in