I would like to embed Flash Player directly inside a C++ or Java application.
I found an article that describes how to do this for C#: http://www.adobe.com/devnet/fl
What you should use is Netscape Plugin API. It's native and it's what all the good browsers use to load flash player plugin. While the actual flash player is not open source, this API is.
If you're not gonna ship a copy of flashplayer with your app:
Your app should check at runtime for flash player and should alert the user if it doesn't find it (usually by giving a link to Adobe to download it).
Flashplayer plugin is usually installed at %AppData%\Mozilla\plugins
on Windows.