I've heard of it being done using Runtime.exec(...) by a person in the java forums, you'll have to google it. I haven't heard of a flash API for java.
Actually, I think you could use the JDIC API to play Flash. I just tried the JDIC browser test provided on their site and it opened the flash animation on http://www.java.com
without any problems! Wow, that opens up a lot of possibilities!
Here's the browser test webstart link: http://javadesktop.org/jdic/demo/Browser/browser.jnlp
And its on the JDIC homepage: https://jdic.dev.java.net/
Good luck using JDIC though, I'm still trying to build it
. Its not as user friendly as JOGL.