I have fixed my exchange push problem. I can't tell you why it worked, only that it did. I had the issue on all IOS devices we have. Desktop mail clients were not affected.
I went to the control panel of the mail server and changed passwords for all users. I then deleted the exchange accounts on each device and did a hard reset. I'm not completely sure that was necessary, but I did it anyway. I then created the accounts again using the new passwords. All of them worked perfectly the first time.
Also, the Automatic Reply is no longer on the accounts. We don't use that function, but it showed up on each account before I deleted them anyway. It showed Loading as it's status. Now, it isn't even there.
Rackspace hosts my mail. I emailed the support people to tell them what I did and they reported back that this is fixing the problem for their users. They also reported that the problem seems to affect a small number of users. I don't know if this will work on all exchange servers. I hope it does.