I'm having similar problems to some of the (old) posts above on my Galaxy Note.
To my great surprise I marginally prefer the Touchwiz email app to the gmail app - I don't do much labelling etc and I like the unread count that defaults on the default email app. I can set it up fine with IMAP but ideally I'd like to enable push with the Exchange settings.
Anyone know how? I tried stevehy's helpful suggestions but alas, didn't work.
I figure the settings will be largely the same on the Note as they are on the S2.