We need a public section to our site and a private section. For the private section we would like to assign a username and password for each user that is NOT their Wordpress username/password. Is this possible?
Yes, there are several possible ways to do this. The easiest method is probably to use Basic HTTP Authentication using a .htaccess and .htpasswd file. Here's a site that will generate these for you. You can create separate usernames/passwords and then drop these files in the private folder. There are several guides and tutorials for this including here, here, and here.