I don't see any problem using other providers beside Gmail with Browserbox as there is nothing Gmail specific used. Browserbox implements plain IMAP4rev1 protocol and only uses extensions if these are available (MOVE, IDLE etc). For example Yahoo does not support IDLE and Browserbox uses NOOP polling instead.
You can see this site to check what kind of extensions are supported by major IMAP providers:
http://capa.kreata.ee/ Browserbox does not assume any of the extensions to be present by default.
Best regards,
Andris