Is it possible to create games with PlayN (in the HTML, Java and Android version) that the user can play via text messages? I'm thinking about games like Tic-Tac-Toe.
Or should I try something different than PlayN to create my games?
There's no built-in support for sending SMS via PlayN. Support for receiving and processing SMS messages would be impossible on non-mobile platforms, and I'm not sure it's possible on mobile platforms (due to a lack of an API). I've never heard of a game sending and receiving data via SMS, but perhaps such a thing exists.
You should consider using this fancy thing we have called the Internet to send your game data. It works pretty well for hundreds of thousands of existing desktop, HTML, Flash, Android and iOS multiplayer games.