STILL IN PROGRESS. These instructions do not appear to be valid as the accounts are now added through account settings rather than in the Gmail App itself.
On the latest version of the Gmail app, you can read and send mail from all your email addresses—even if they're not an @gmail or a Google for Work account.
Add an account
...
...
...
...
...
...
...
If you run into problems adding a non-Gmail address, you can check for other setup instructions on your email provider’s support site. When looking for this information, you can do a Google search for the words "POP" or "IMAP" along with the name of your email provider.
...
...
...
...