This is a stumper... helping someone that can't get into any secure site at all.
I've tried an "everything including the kitchen sink" approach, including:
Winsock reset utilities
Reset IE to default
CCleaner - let it clean everything, including cookies
Resetting DSL modem and router, and trying both on a separate machine - my laptop - which logged in just fine.
I've updated the NIC card drivers, too.
I ran the Symantec Removal tool, in an attempt to get rid of any possible lingering traces of Norton Firewall.
In IE, I made sure SSL 2.0 and 3.0 were checked.