Have you considered adding user roles to the page so that you need to log in to view the page? That would be the best way to protect the page. Even if you do manage to hide it from internal search, outside crawlers can still find and index it.

You could also look at adding the search-boost module to the site. That module replaces core search and allows you to exclude specific pages from internal search.

Keep in mind that even if you use a third party module like search boost, you will still want to add Disallow: /yourprivatepage/ to the robots.txt file of your site as well to ensure it is not indexed by search engines.