Symtoms: Clicking "Add" after selecting an attachment simply refreshes the screen with all fields cleared. This problem can also manifest itself when clicking any button from the Compose screen results in simply refreshing the screen.
Problem: PHP is not enabling uploads.
Solution: Edit /etc/php.ini. In the section labelled File Uploads change file_uploads = Off to file_uploads = On.
Or, if you don't have access to /etc/php.ini (such as in a shared hosting environment), create a file in the SquirrelMail directory called .htaccess with the line php_flag file_uploads on.