Web-based mail services do not store mail files on disk, so there is nothing for EagleFiler to read. The best way to import Webmail is to connect to the mail account using one of the above mail programs. If your Webmail does not support this, you can drag and drop text from your Web browser, use File ‣ Save to save messages into EagleFiler’s To Import folder, or print and choose Save PDF to EagleFiler.
You can also import individual messages by saving their raw source to a .eml file. This has the advantage of preserving the message’s attachments. In Gmail, this is possible with the Show Original command.