You can use the Reshape operation to generate additional attributes and then export the resulting scratch layer.
the_geom=the_geom
FIPS_CNTRY=FIPS_CNTRY
GMI_CNTRY=GMI_CNTRY
ISO_2DIGIT=ISO_2DIGIT
ISO_3DIGIT=ISO_3DIGIT
CNTRY_NAME=CNTRY_NAME
LONG_NAME=LONG_NAME
SOVEREIGN=SOVEREIGN
POP_CNTRY=POP_CNTRY
CURR_TYPE=CURR_TYPE
CURR_CODE=CURR_CODE
LANDLOCKED=LANDLOCKED
SQKM=SQKM
SQMI=SQMI
COLOR_MAP=COLOR_MAP
the_geom=the_geom
FIPS_CNTRY=FIPS_CNTRY
GMI_CNTRY=GMI_CNTRY
ISO_2DIGIT=ISO_2DIGIT
ISO_3DIGIT=ISO_3DIGIT
CNTRY_NAME=CNTRY_NAME
LONG_NAME=LONG_NAME
SOVEREIGN=SOVEREIGN
POP_CNTRY=POP_CNTRY
CURR_TYPE=CURR_TYPE
CURR_CODE=CURR_CODE
LANDLOCKED=LANDLOCKED
SQKM=SQKM
SQMI=SQMI
COLOR_MAP=COLOR_MAP
DENSITY=POP_CNTRY/SQKM
You can define additional columns using any CQL expression, please be advised that the shapefile format can only support one geometry column.