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JOGL (and OpenGL) doesn’t have any specific file format tied to it. There are certain ways that make sense to store geometry when using different rendering mechanisms. For example, if you’re using vertex arrays or vbos for performance, you’ll likely find you’ll have to do some work to get 3d formats into a workable state.
At the moment, I don’t know of any blender file loader. However, blender has many export scripts and I believe some engines and libraries exist to load those. Also, it is possible to write your own blender export script. I’ve done this before and it wasn’t too difficult.
As for what you need for board games, that all depends on how good looking your games need to be. JOGL handles rendering things; if you can think of a way to describe in combinations of points, lines, and triangles/quads (including textures, custom glsl shaders, and particle effects) then JOGL will work fine. I can imagine that many of the advanced features of opengl won’t be needed by you.
Also, as a friendly recommendation, if after you’ve learned enough of opengl to get comfortable, I would suggest using a graphics engine built around jogl (or similar) to make designing the board games much easier. Some possible examples include jMonkeyEngine, Xith3d, and Java3d. This approach is much better than designing your graphics system.
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You can use JOGL with JMonkeyEngine 2.0, I have begun to repair the JOGL renderer used inside this engine and it works quite fine. If you want, I can explain to you how to start, let me know if you’re interested.
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If your games are tiny, maybe JOGL is enough. If you want to do some more, for example loading some models, you can use JOGL and JOGL-utils. If this is not enough and if you want to avoid reinventing the wheel, use JMonkeyEngine. I used only JOGL for 2 years and when I began writing something that was like a scenegraph, I wondered if it would be better to use an existing one rather than rewriting an worse equivalent of JME.
JMonkeyEngine is a 3D engine, not a game engine but it helps a lot, there are many features, you can use it to load many formats of models and it is a quite good scenegraph, you can use it to improve your performance. It is an higher-level solution than only an OpenGL binding, it allows you to save a lot of time to concentrate more on the gameplay once you know a bit how it works. I think that it is easy to succeed in doing something with JMonkeyEngine if you have some time to study the existing example. The documentation is sometimes a bit weak 
Nevertheless, you need to use Java Webstart or another installer (IzPack for example) to allow people to launch games using it. The casual users won’t install JME separately.