I'm looking for smallest linux distro with a GUI that natively runs JRE. DSL doesn't have JRE and I don't really want to spend 15 hours trying to figure out how to install it. I'm looking for something that pretty much works out of the box.
The JRE comes as an RPM or Deb installer. In reality though they are simply archives that go into the package manager. So theoretically you should be able just to extract the contents of one of the installers, dump it in your filesystem, and resolve any missing library issues.
However doing this is a little crazy. Why not take Ubuntu server, install X, then just run the JRE in there? No window manager, no desktop environment, just your Java program. That's plenty light enough if you don't want a DE.