Sending an email is a server side action. Breakout is a client-side JavaScript library so you’d need to send an ajax request or post a form to a server and the server would actually send the email.

You could send an email directly from an Arduino if you have an ethernet shield or wifi shield. However you’d have to write your application in c/c++ and have the required shield.

The best option for what you want to do is to use johnny-five. It’s similar to Breakout, but it’s server-side (nodeJS) so you can send email directly. See this article on using johnny-five to achieve exactly what you are trying to do: http://blog.sendgrid.com/how-to-send-email-with-arduino-at-ny-tech-meetup/.