Seeking GeoJSON data for all counties of UK?

A simple approach is to make one if you can't find one. Get the free OrdnanceSurvey OpenData Boundaries dataset. This contains several boundary datasets in either Shapefile or MapInfo formats. Identify the one you need and load it into a competent GIS (QGIS is free). Then export as GeoJson. Alternatively use org2ogr (also free) to convert it on the commandline.

EDIT:
To get a full set of the county boundaries for Great Britain (not UK) from the Boundary Line data, open 'District_borough_unitary_regions' and then dissolve by 'FILE_NAME'. In a few instances (Leicester being one) the county-city will have a separate boundary (being a unitary authority in its own right). Dissolving with a reg-ex expression, or using a look-up table of counties and county-cities and some judicious table joining will sort that out.

Alternatively...
You should be able to extract them from OSM data (e.g. Geofabrik). You will need to filter the polygons to extract just the administrative boundarie. According to the documentation, you need Admin level 6 (administrative counties / Unitary authorities, City of London)... but since this is both unitary authorities AND counties you have the exact same problem as above (which is not surprising since OSM uses the OS Open Data). A quick visual check confirms the OSM Admin-level 6 boundaries and the OS open data boundaries are almost identical. The one advantage of this is you can get the whole of the UK from OSM.

I would download data from here if nothing works and convert shapefiles into geojson.

I ended up using RafealJS with the following SVG file: http://commons.wikimedia.org/wiki/Category:SVG_maps_of_the_United_Kingdom#mediaviewer/File:British_Isles_in_counties.svg

Would the Natural Earth Admin 1 data work? http://www.naturalearthdata.com/downloads/10m-cultural-vectors/10m-admin-1-states-provinces/ (again, you'd need too convert from shp to json)

I would download data from here https://geojson.co/ , if you need all the countries in one file, then input - England+Scotland+Wales+Northern Ireland, unless you need each county in a separate file, then you just input each country name one after the other download.