Time to bump this up and add some more information.
So far, I've tried Monal, IM+, IMO, munduIM, Talkonaut, and BeeJive for Jabber. None of these will allow me to login to our server. I can't even find decent application logs and nothing shows up in our server logs.
What I don't understand is how Push Notifications in iOS are based on XMPP, but there's no decent chat implementation of that!! Messages is great for iOS. Can I add a private account to that application, please? Apple? Please?
Does anybody else know why on earth these apps won't do the basic thing that they advertise?
You almost certainly have a misconfigured router/firewall for jabber. I can confirm that OS X Lion Server works with the iPad and iPhone IM apps imo and BeeJiveIM.
To track down your problem, which almost certainly is simply a port getting blocked or not forwarded at some point between your router and server, I'd suggest getting an nmap executable (e.g., get Xcode via app store, then macports, then sudo port install nmap) and download the old feature-rich Airport Utility 5.6 from Apple. First try this:
$ nmap -p 5222 localhost
$ nmap -p 5222 server_ip_on_LAN
$ nmap -p 5222 server_ip_on_INTERNET
This port should be open for each probe. When I first set up my server, Messages beta added a port forwarding entry on my Airport router, but this had a duplicate entry with another port forward, so I had to use Airport Utility 5.6 to track down and delete the superfluous port forward entry. I also had to use Server Admin to open all the iChat ports.
One thing I was hoping for is an iOS app that supports video/voice over jabber, but all the apps I see are strictly IM. Can anyone recommend an app for this?