I don't click the button twice. The same happens when I press the Enter key.
The NagAuthn application is extremely complex. However authentication is done using the OAM API and the OID counter I believe is either set manually through an LDAP update or through a connection to OAM, in which case the OAM authentication is being called twice. It looks as though the logged out behaviour of OAM is causing some issues - possibly with IE not voiding some cookies. PM me if you like.