Firstly, you have to specify what you mean by "native" code for the different platforms.
On Android, your java files are specifically compiled/prepared for dalvik. So they are already "native" of a fashion, no work needed to be done here. If you want to get C/C++ native code for Android using the NDK, you're out of luck. PlayN doesn't do this and this is a hard problem (going from Java to C++)
If you take a look at the Maven modular layout of how PlayN is intended to be used, it isn't difficult to define a Factory interface in the common code and pass in a platform specific implementation for each module. It's no big deal to support Android specific functionality this way.
For the HTML version, you can use HTML libraries no problem using JNI, although really garnering specific functionality of the browser I'd imagined of limited value compared to what PlayN has already exposed. The one thing that is useful is text/keyboard input, although I'd recommend triplePlay https://github.com/threerings/tripleplay UI library as they've solved this, and it's an active project.
As for iOS, this might be more complicated as the iOS module is a bit of a hack where the compiled Java classes are run through an JVM runtime for .net (IKVM) and then uses the Monotouch tools to compile the whole thing to native code for iOS. See https://github.com/samskivert/ikvm-monotouch
So for iOS, you won't be able to bind the code to any form of native version, and what you have access to via the toolchain method depends very much on what Monotouch has catered for iOS (quite a lot I imagine), and also what IKVM-Monotouch has supported (I imagine the bare minimum to get PlayN working).
I'm not familiar enough with the Flash pipeline to give you an appraisal, although I think that it's quite flexible.
The above answer is written assuming your app is actually a game. If it is not and you intend to use the standard widget libraries for various platforms on mass, it should be possible. Choosing a good MVP framework would be good here, and depending on the assumptions it makes on different host environments will determine how easy the whole thing will be.
I'd recommend reading and comparing https://developers.google.com/web-toolkit/articles/mvp-architecture and perhaps look at questions like What is your favorite GWT MVP Framework?
...although a lot of these frameworks might be GWT specific and not really have catered to being reused on other platforms.