A secure text messaging app I like and use is Threema (http://threema.ch/en/index.html). All servers are allegedly located in switzerland. The app uses public-private-key end-to-end encryption.
You can exchange keys locally via QR code and it is available for both Android and iOS.
You can still sync your phonebook in whatsapp style. It's a good compromise between being secure and being pragmatic to me.
However, it's not open source, so you still have to trust the app (and the OS) not to send out your private key.
If you're jailbroken (which you probably should consider if privacy is a concern), FirewallIP is the reason I still haven't switched to Android: http://yllier.webs.com/firewall.html
Can you expand on why one should jailbreak? Any other jailbroken app suggestions? I was under the impression that jailbreaking also opened you up to be hacked and made the phone less secure.
That's unfortunately the problem. We want to make secure software easier to use but we can't get past step 1 if people can't just buy it from the app store.
ChatSecure and IM+ both support OTR instant-messaging on the iphone, so depending on what you mean by "like that"... yes. ChatSecure is opensource too fwiw: https://github.com/chrisballinger/Off-the-Record-iOS
https://www.mywickr.com
https://mike.tig.as/onionbrowser
https://www.getcloak.com
Silent Circle as well, closed source or not, given that one of the founders is Phil Zimmerman.