I'd like to add functionality to the existing iOS Mail app, specifically - the ability to hide emails for later review (e.g. 'snooze' functionality). I know this functionality existing is multiple third party mail apps like Mailboxapp etc but all of these third party mail apps for iOS only work with Gmail and I want to use this 'snooze email' functionality for non-Gmail accounts.
As such, I can't find the Apple API to modify the mail app to add this 'snooze email' functionality.
How would I go about adding functionality to the existing iOS mail app?
This is an interesting question, but at first glance it seems simple. Remember the rule of thumb? You can't modify apps that aren't your own.
As such, I can't find the Apple API to modify the mail app to add this 'snooze email' functionality.
You can't find the Apple API because there is no such API.
You do have three other options:
File a radar. Perhaps Apple will add this in an OS update if you ask nicely. You do have to ask, though.
Jailbreak and roll it yourself. You'll have to figure out what API exist all by yourself, without any Apple documentation to hold your hand. Your best bet is to try running otool or class-dump on the iOS frameworks to see if there are private methods you can use to set the status of an email. Once you have that working, you can try to figure out how and where you want to build out the user interface or gesture to "snooze" an email. It goes without saying that you can't distribute OS plugins on the App Store. If you want to distribute this, write your own email client and include a snooze feature.
File an enhancement request with other companies, such as Sparrow, Mailbox, and Google, and ask them to add this to their own apps. As you note in your question, Mailbox includes this already. You can try Sparrow and see if it supports snoozing (I don't know if it does non-gmail accounts.) You can also request that they support other kinds of email accounts.
If you're just looking to remind yourself to reply to emails, you can also schedule reminders in the reminders app or via Siri.
Actually, I just flag emails. There's a flagged inbox in iOS 7.