I believe JWebPane is going to be the official way to embed a browser into a java app. Its based on the open sourced engine - WebKit, which is used in Apples Safari and Googles Chrome browsers.See this blog for details.
You could use SWT for your GUI. Its Browser control allows you to embed IE, Mozilla or Safari (depending on the platform you're running in) with little pain.
By far the most robust embeddable browser I am familiar with is the one in SWT.
In fact, it is so flexible that the JavaDoc hover you can see in Eclipse is actually a browser, and the JavaDoc view actually supports things like animation!
The only risk with using SWT is that there are different versions of the SWT library for different platforms. I'm not sure if there is a singl jar you could include to cover everyone.
If you need a pure Java solution then you can try JWebEngine. It render HTML 4 very good. You can use it in an applet, Java webstart and on any platform. The using is very simple.
You could try a JEditorPane, it doesn't interpret advanced HTML, nor Javascript, nor advanced CSS, but you can write that part yourself, called the EditorKit. That is the class/object that is consulted by the JEditorPane or how it has to display its content.
I know its possible, because I tried and failed (:P), but it could be outdated or deprecated by now, I don't know.
If you look at The Minecraft launcher (the old one), look through LoginForm or LauncherFrame, you may be able to find out that method. There is a tutorial by kippykip on youtube on how to decompile and edit it: here