You cannot get push email with Juice Defender... I have both imap idle push accounts via the gmail app, and a work email via exchange active on my phone... Juice Defender saves battery by limiting up time on the data connection... you may think you have push if you've customized settings to something like 2m every 5m for data, but it's not true push... for most of us, that's damn near close enough though, and if it saves even a little battery, probably worth the tradeoff... I am personally set at 15M / 5M and it seems to deliver emails pretty efficiently... if you think about it, and assume that incoming emails are staggered over the day, that setting should get emails to you generally within a few minutes of when they were sent, sometimes sooner...
The only way juice defender would allow push, would be if you disabled it for managing data, and only used it to manage wifi, night schedules, and things of that nature... your email programs do not overcome juice defender's settings as far a I have experienced, and I've monitored this for a while now...