This was done on SharePoint 2010 (not sure how), but I am on SharePoint Foundation 2013 and I need to have this site as a link on the top link navigation so a small group can access it, but I want to hide it from all other users. Since I have Foundation I do not have target audience and I do not have Navigation under site settings. In 2010 I know it was done because I can see it, but when I sign in as someone else I cannot see it on the link bar or in site contents.
What you are describing is the native functionality of SharePoint. What you need to do is for that secure site - set up corresponding security groups and permissions, then go ahead and add a link to the top link bar. That's it. The link will only be seen by those who have access to that site and no-one else.
Everything in SharePoint is permission-driven, which means that you will not see the link in the navigation or content in search results if you do not have access to that site. You do not need publishing features or target audience settings for this to work.