The account type you set up your account with is almost certainly POP. With POP you only see Inbox. In order to see the other folders you'll need to set up your account as IMAP, and subscribe to the folders you want to see in Thunderbird. Check with Verizon about the correct server settings for IMAP.

Note, you cannot convert an existing account. See http://kb.mozillazine.org/Convert_a_POP_account_to_a_IMAP_account

Thanks for your response. I have searched Verizons website and forums and apparently Verizon does not support IMAP, they only use POP3 servers. Any other suggestions on how I might download the messages on the Webmail My Folders?

Put the messages in your inbox on Verizons webmail. I suggest one folder at a time so the mail does not get mixed up, then move it when it gets to Thunderbird.

Thank you. I had actually worked this solution out and found it worked great. Since I have multiple Thunderbird acounts on different computers, I did get a number of duplicate messages, but these can be worked out. Thank you again.