Restricting access to specific uploads

So I am making this specific private page that logged in users will find downloadable files. BUT I don’t want the uploaded media to be accessable through attachment links. They should also be private.

Is there a fast and easy way to do this? I just want something like a checkbox to make a uploaded media file private. There will be many files that should NOT be private.

This needs to be finished today, on a tight schedule here! Any help will be appreciated!