I'm unclear on what your final result is supposed to be...Will the final shapefile be of polygons that match the polygons in the census shapefile, but they will also contain attributes from the county shapefile? Or will the final shapefile be of polygons that don't match either of your shapefiles and will be created from scratch?
Have you looked at the 'Split' tool (Analysis Tools > Extract > Split)? This will split up your county shapefile according to overlapping areas in your census shapefile.
The other possibility is to use the 'Identity' tool (Analysis Tools > Overlay > Identity). I'm not sure about this one, but read the 'Tool Help' when you open the tool and see what you think.
Hope this helps.