This is tested to work with your data and GDAL command line tools installed from OSGeo4W on Windows.
Test your data with ogrinfo. Escaped double quotes \"\" are needed because the layer name contains dot character.
ogrinfo -dialect sqlite -sql "select * from \"ks_kansas_zip_codes_geo.min\" where ST_IsValid(geometry)=0" ks_kansas_zip_codes_geo.min.json
Confirmed that there is one error
GEOS warning: Self-intersection at or near point -98.918045000000006 37.379078
Feature Count: 1
Create a new, fixed GeoJSON file by utilizing the ST_MakeValid function of SpatiaLite http://www.gaia-gis.it/gaia-sins/spatialite-sql-latest.html
ogr2ogr -f GeoJSON -dialect sqlite -sql "select ST_MakeValid(geometry),STATEFP10,ZCTA5CE10,GEOID10,CLASSFP10,MTFCC10,FUNCSTAT10,ALAND10,AWATER10,INTPTLAT10,INTPTLON10,PARTFLG10 from \"ks_kansas_zip_codes_geo.min\"" clean.json ks_kansas_zip_codes_geo.min.json
Check the new GeoJSON file. Ogrinfo has showed that the layer name inside GeoJSON is SELECT which is reserved keyword for SQL so it needs to be escaped.
INFO: Open of `clean.json'
using driver `GeoJSON' successful.
Layer name: SELECT
Geometry: Unknown (any)
Feature Count: 0
You must have GDAL that is built with new enough SpatiaLite which has the ST_MakeValid function. If you do not have such version you will see this error:
ERROR 1: In ExecuteSQL(): sqlite3_prepare_v2(select ST_MakeValid(geometry),STATEFP10,ZCTA5CE10,GEOID
10,CLASSFP10,MTFCC10,FUNCSTAT10,ALAND10,AWATER10,INTPTLAT10,INTPTLON10,PARTFLG10 from "ks_kansas_zip
no such function: ST_MakeValid