This is a general answer, ie not specific to Verizon e-mail software: did you at any time change your e-mail settings? Some systems save them automatically and your option is to stop them from doing so; others don't but provide an option for you to do so - really depends on total mailbox memory available. All e-mail is saved somewhere on the sending and receiving servers as well as on intermediate nodes - in addition to your own system, if applicable - so theoretically you can always get them back. If you think they're vital for some reason - proving a legal case, eg - you can ask Verizon to retrieve them; they can do it and are legally obligated to do so, but you may have to file a police report and/or obtain a court order first.
For future messages you'd like to save, simply send a bcc to yourself - then you can always save that in a different computer file.