The trouble with searching for info on this specific subject is that if your search includes results from more than one year back then you might be reading something that is completely erroneous now.
I think it was the end of January 2013 when Google stopped allowing new devices to have Gmail or Google Apps Free set up as "Exchange". I'm not sure of the current status of "grandfathered" devices as it has been a long time since I've tried it with one.
So...if you have one of the above type accounts you'll need to set up as "Gmail" on the stock Mail app and will not have push. Google's Gmail app will receive push notifications, however.
A jailbroken device can use the tweak "Push for Gmail" to get an effective push in the stock iOS app.