Amex uses the same log in as you use to log into the web site (at least mine does). Make sure you can log into the American express web site.  What version number of QuickBooks Mac do you have? 

Select your amex account in the QuickBooks Chart of Accounts window. Choose Edit account. Click on Online settings. Choose disassociate.  Now go to banking->Banking set up and set it up again with your log in user name and password (the one that works on the web site).