If your old hosting plan uses cPanel and your HostGator account has cPanel*, we can accept a Full Backup file, which contains any emails on the old server. We can then restore those on your HostGator server.
If your old and new hosting plan does not use cPanel, we cannot promise the emails will transfer.
If your old hosting plan uses Maildir, you may email transfers@hostgator.com to inquire if moving the emails is possible.
(*All our shared hosting, reseller hosting and Linux dedicated servers come with cPanel. VPS can choose cPanel or Plesk, and Windows uses Plesk.)
Ok, so it's not possible. Now what?
We suggest you use a POP3 connection (via Outlook or Thunderbird) to archive your emails. Then the emails will be safely stored on your computer. However, they still cannot be uploaded to the HostGator server.