Thunderbird makes a secure connection to your e-mail service provider via IMAP such as to Gmail. As does Eudora and Outlook, though Outlook prefers pop3 instead of IMAP (I am not sure how Eudora works natively as I haven't used it before). Though that secure connection is relative and can always be hacked if someone wants to take the time to target you specifically (all security is relative in that matter and generally only keeps the good people out). However, after the connection to the e-mail service provider the e-mail is only as secure as the that provider is secure. If you have a yahoo address or hotmail address (or generally most e-mail providers along with anyone using a standard exchange server) the e-mails are then sent unencrypted via the internet to their destination.
I don't know about Eudora (but I am sure it has this) but both Thunderbird and Outlook are able to use some sort of PGP which a way for encrypting e-mails. However, this is a difficult process and your recipient needs to have PGP also setup properly so that you both have each others public keys and are able to unlock the messages received and private keys so that you are able to properly lock messages that you want to send. This is not an option for the average user and to send out encrypted e-mails to everyone. It does however work well in a corporate environment where there is IT support to make sure all senders and receivers within the corporation are sending and receiving secured e-mail. However, once you e-mail someone outside the company who doesn't have your public key then you need to make sure you send an unencrypted e-mail.
Firefox is a browser and not an e-mail application (i.e. Eudora, Thunderbird, or Outlook) and is not an e-mail service provider (i.e. yahoo, hotmail, gmail). Using Firefox has nothing to do with e-mail security.