Cannot connect to webmail, or popmail

Question

This problem began about 2/11/15. I didn't notice it right away because access from work is working fine. Access from home is the problem.
We have had the same setup, FIOS, modem/router/coputers for at keast 2 years. No changes have been made. I still use eudora on my computer, my wife uses incredimail on hers. Both of us have the same exact problem.
From the email client all attempts to rec or send mail return a "cannot connect to server" response.
I can access the verizon web site with firefox, scroll to the bottom and select check mail, and the page times out. Entering the address webmail.verizon.net also times out with a page saying auth.verizon.net has timed out.
On any and all verizon web pages that have any kind of sign in button, I cannot get it to work. It usually comes up with a blank box, or nothing at all.
I have tried firefox and iexplorer.
On Sunday, a Verizon tech spent about an hour on my computer, installed chrome with the same result, reset the router and the fios box on the outside of the house. He tried pings to webmail.verizon.net and they timed out. He finally said it must be on their end. This morning I contacted them from work because it still wasn't working at home. They wanted to run me through all of the same steps the first tech did on Sunday.
I have tried using an alternate DNS with no luck.
*Everything* else works. My non Verizon email works, all web sites work including secure sites. If anyone has a clue please advise. I don't know what else to do except spend another hour with the Verizon guys and feel we are getting no where.
Thank you

Answer 1

I read your post when I got home last night but could not reply. When I click the reply button the page timed out with the message:
The connection has timed out
The server at signin.verizon.com is taking too long to respond.
To answer your question, yes. I cannot access my account or anything requiring a "sign in" on the Verizon site. When I go to verizon.net and click the sign in button at the upper right, a blank box, about 3 inches drops down with a dotted outline and completely blank inside.
I also did a trace to pop.verizon.net and will post the results. I am not that savey with this stuff so I'm not sure what it means but I believe it shows I cannot connect to it.
Tracing route to pop.verizon.net [206.46.232.132]
over a maximum of 30 hops:
1 1 ms 1 ms <1 ms Wireless_Broadband_Router.home [192.168.1.1]
2 30 ms 32 ms 29 ms L100.NYCMNY-VFTTP-229.verizon-gni.net [74.101.23
.1]
3 9 ms 7 ms 13 ms G101-0-0-19.NYCMNY-LCR-22.verizon-gni.net [100.4
1.211.210]
4 9 ms 16 ms 9 ms so-3-1-0-0.NY5030-BB-RTR2.verizon-gni.net [130.8
1.199.12]
5 31 ms 11 ms 9 ms 0.xe-1-3-1.XL2.EWR19.ALTER.NET [152.63.6.173]
6 13 ms 9 ms 10 ms 80.xe-11-0-0.SA2.EWR19.ALTER.NET [152.63.24.190]
7 * * * Request timed out.
8 * * * Request timed out.
9 * * * Request timed out.
ditto 10 thru 27
28 * * * Request timed out.
29 * * * Request timed out.
30 * * * Request timed out.
Trace complete.
I dusted off a laptop I haven't used in 5 months, fired it up and immediately went to the verizon site and the sign in timed out. Email also did not work. My home computers use Win 7 Pro, but the laptop is Win XP sp3.
It has been almost a week now that we have not had email service at home and no one at verizon takes this seriously, only wanting to get on my computer, which I did cooperate with, and they found nothing wrong. But there the support stops. I've been a Verizon DSL/Fios customer probably 15 years, but there are alternative services available that work. This is getting frustrating.
Thank you.