If you still have access to the original service, the best method by far is to use the tool ìmapsync (or OfflineIMAP as an alternative).
This will allow you to temporarily sync from the old service to the new one. It will retain all of the flags as well so that unread markers are retained. It will also retain any folder structures.
The second most common way of achieving this would require some careful coordination of mail routing. That would be a file copy of the source data which would normally be either in maildir or mbox format. This may also require help from the previous mail provider unless you have shell access to the old service.
imapsync is certainly the preferred method. Trying to do a transfer using eml files is not recommended. For starters, you will have lost all of the flags and folders. In addition, trying to do this for 5-6GB per user is going to take a LONG time. You will have to do it in stages.
Additionally, I'm not really sure that Pine or MUTT is going to help doing it the way you've outlined although you may be able to write macro's to transfer the files a few at a time.
UPDATE:
As we now know that sync from the original is not possible. The best way to script input from EML files to a maildir based system (if that is what you have, it is the most common storage format for Linux IMAP servers) is to use
getmail_maildir ~/Maildir/ < email_file.eml
getmail_maildir is part of the getmail package. This only works if you have direct access to the mail folders though this is commonly true with the better hosts. Not so sure about doing this with the other mail storage format mbox but I think that getmail also has a getmail_mbox command. In addition, the Windows application "IMAPSize" has a command for converting from EML to mbox.
So again, it is much easier to migrate the emails to a physical mail store rather than trying to pass everything through IMAP. However, it may be that you have to do this because the new provider cannot provide suitable access (as would be the case if migrating to GMail for example).
If this is the case, what may be best is to migrate the EML files to maildir format using a "synthetic" local maildir (there is nothing really special about maildir except the naming conventions so you don't actually need an IMAP server to use them) and getmail_maildir. Then use IMAPSync or OfflineIMAP to push from that local maildir to the new service. That way you don't need to mess with trying to script MUTT.