Here are the usual reasons for this to happen:
1) The "register.php" file is not in the "affiliates" directory of the plugin (maybe it didn't get uploaded?)
2) The "register.php" file has wrong file permission setup (The proper permission of that file should be 644. If you have loose permission settings on that file then your server will block that script from running it due to security issues)
3) You have a .htaccess file issue (e.g. some line in your .htaccess file is restricting the "register.php" file to run)
The following two links have two examples of this .htaccess issues:
http://www.tipsandtricks-hq.com/forum/topic/link-for-new-affiliate-registration-goes-to-404pg
4) A memory limit on your server is stopping the PHP script (register.php) from executing. If you contact your hosting and tell them that executing a PHP script is giving a 404 error then they will know what to do.
Does any of the above conditions apply in your case?