I was having problems with this all day. The way I fixed it was by going into my router settings and configuring the firewall so that the Everquest client would specifically be allowed access.
For those of you using AT&T Uverse, here are instructions how to do this:
1) Enter
http://192.168.1.254 into your browser.
2) Go to View Firewall Summary
3) Click Firewall Settings
4) Make sure "Allow individual application(s)" is selected and scroll down the list of programs and add Everquest to the Hosted Applications.
I haven't had a problem logging on since. Now if only I could zone without getting kicked off. =/