After suffering like many of you, and not being able to access Verizon e-mail from Thunderbird (also relevant to any of you that use Outlook or other non-web based mail programs) finally got solution very quickly once a good Verizon USA based Tech focused on my problem, issued Verizon Repair Ticket number and problem was solved from Verizon's end in less than 1 hour.
So, the solution was simple. I provided the IP Address where I am currently overseas. Verizon then "white-listed" this IP address, where before it was "black-listed". Here are some practical suggestions if you're having similar problems.
1. If you reach Verizon Tech Support in any country outside USA ask to be transferred to a USA Tech. Initially, I was connected to multiple techs in Mexico that said that a Verizon Repair Ticket could not be issued for this problem. That statement is not true; Repair Tickets can and were issued for this problem. Insist that a Repair Ticket be issued for your problem. Don't accept anything less!
2. Verizon techs in Mexico admitted that they don't have access to same level of information that USA techs do. Mexico said there was no record of this collective problem that many of us have experienced since mid-September. I told them to have a look at Verizon e-mail forums!
3. No one outside of Verizon can solve this problem as the problem exists on Verizon's mail servers.
4. Problem first started on or about September 17, 2014.
5. When you deal with Verizon Tech Support in the USA be prepared to give them the IP address where your computer is connected where you currently can't retrieve e-mail. It is easy to get your IP Address, just Google "what is my IP address" and you'll get a series of numbers, something like 123.45.678.901. Give this to the tech so that IP address can be white-listed.
6. As a practical matter if you're switching locations where you access e-mail, you may have to go through this procedure for each location where you want to get e-mail. Just depends on whether that location is black listed or white listed. This suggestion may not be practical if you're traveling to multiple locations overseas and are not in any one location for more than a couple of days.
As an aside, I think it is unconscionable that Verizon doesn't post a summary of this problem and the solution(s) for it. Hopefully this posting will save you from making multiple 1+ hour phone calls to Verizon as I've done in the last 3+ weeks.
Dear Wayward American,
Thanks for your optimistic post. But for the average overseas traveler who is on business or on a touring vacation and is not at one place for more than a day or two, your comment "This suggestion may not be practical if you're traveling to multiple locations overseas and are not in any one location for more than a couple of days." renders this "solution" by Verizon quite useless.
Many overseas hotels have pretty dodgy e-mail systems anyway and may well give you a different IP address every 24 hours!