I just checked, your link works in IE7. Notice that the page you link isn't secure, but the page after it (where it takes your CC info) is secured. Probably your bill-pay sites are secured too, which leads me to think that it might be a problem with the way your copy of IE6 handles secure certificates.
I did a little googling for you, and found that this is one of the bugs Service Pack 1 for IE6 fixes:
"Internet Explorer also may not show a client certificate when you try to connect to a secure Web site after you are prompted for authentication by the secure site.... In some cases, you might receive a "Page cannot be displayed" error message when you try to connect to the secure Web site. "
You can download SP1
here.
I doubt that's the only patch IE6 has for certificate issues, so you might want to do some more googling. Trying Windows Update will, unfortunately, probably hide all IE6 updates and try to bully you into upgrading to IE7 (which I recommend, but a lot of people don't.)
Edit: Errr, missed something--the patch I linked isn't actually SP1, it's a patch *for* a bug *in* SP1. So you'll need IE6SP1 to use the patch.