One way to juice really fiberous stuff is to fill a blender with water, blend the stuff, strain it in a potato ricer (handheld press) and then put the liquid back in the blender with some more vegetable matter.
After the last press, pour through a fine mesh filter to get the smaller particles and then drink.
I own a champion juicer, but I end up using my blender instead for daily juices because a blender is a lot easier to clean.
RSVP International has the best ricer I've found for this, with a small mesh that locks nicely in place and can be removed for easy washing.