UPDATE -
I just got an idea - I logged onto www.att.com on my iPhone using Safari, and searched for the "FULL WEBSITE" button at the bottom of the page. I was able to log onto my ATT account using this "full website" option, so this will help me until AT&T comes up with a fix.
I still can NOT logon to www.attcom on my iPhone using Safari with the MOBILE site, nor can I logon to the ATT app on my iPhone.
I looked at the ATT app info and I see that there was an UPDATE to the ATT app around May 9, exactly when I started having my problem. I guess there is some bug in the new update ATT app, and it is probably in the MOBILE web program, too.