(Well, the Gmail iOS app certainly doesn't; I haven't tried the Outlook client but the description in the app store suggests a very limited set of providers.)
Last I checked, the gmail app only worked with gmail accounts.
Of course, to trust that feature, it also has to be tremendously reliable, and that's the biggest thing that Mailbox got right in the first place. That's also why it very handily beat out Mail Pilot, which had the same idea and got started earlier but to this day is still filled with terrible bugs.
Email should be simple but there has been essentially no lasting innovation in email clients in a very long time unless you really want Google to host it for you
Outlook (incidentally, another example of my original complaint, as it's just former email startup Acompli with a fresh skin although obviously it's probably not getting shut down anytime too soon) doesn't have very good message searching for Gmail, which is a common complaint I have for non-Google mail apps. Obviously, I get that Google has a huge built-in advantage here; but if your email app is going to support the "Mailbox paradigm" of abandoning folders/traditional labels in favor of just marking things as done, it also needs to have search functionality that can back it up. Outlook was also really buggy for me when I last tried it, although that was a little while ago so it may have improved.
My favorite tiny detail is that I can tune the notifications to push only unread counts to the icon badge and only for the "focused" inbox. That means I can avoid constant push notices for email yet still be able to glance down when I want to and see if I have any unread emails that are actual human communications.
http://techcrunch.com/2015/03/30/gmail-for-android-gets-a-un...
Instead I get downvoted. Gee, thanks. Since when has downvote come to mean disagree on HN?
Just curious, how is processing hundreds of emails in Mailbox or Inbox better than Outlook? Seems like trying to do that on a mobile device would be hard no matter what app.