Soln1: http://sourceforge.net/projects/junit-addons
Look at that. It has a PrivateAccessor class to call private methods and fields. It also has utility classes for array comparision and the likes. I really like it.
Soln2: Make it non-private. Or for every private foo(), add another method foo_FOR_TESTING_ONLY() that calls foo(). Or zap the class with reflection to make the method non-private temporarily.