As of August 2016 GeoJSON is now a formal IETF specification. And some things have changed from the old informal 2008 spec. So to be valid for the 2016 spec your polygons MUST be right-hand wound.
See https://www.rfc-editor.org/rfc/rfc7946#section-3.1.6
The use of MUST indicates an absolute requirement of the specification.
However then the subsequent language about not rejecting other windings is weird and is open to some argument - perhaps parsers should accept and correct other windings. I think this what lead geojsonlint (from geojsonhint) to use the word should instead of MUST in their feedback message.
So I agree with the position of geojsonlint.com as they place right at the top of their page the link to the 2016 spec. Your polygons are bad per the 2016 spec. But yet they are fine per the informal 2008 spec. So if you do not wish to change the winding of your polygons, you will need to make certain any linters or parsers you use honor the old 2008 spec and do not one day suddenly migrate to the new formal spec.
Or perhaps you should consider changing your polygon winding as doing so will get you in line with both specs and better positioned for the future as the 2008 spec fades away.
Note that for me the largest change in the 2016 spec was the dropping utterly of the support for coordinate systems. I had all my GeoJSON in NAD83 and then had to quietly remove that as now all GeoJSON is WGS84. Fortunately for me coming from Oracle Spatial all my polygons were already right-hand wound.