It might be easier to use
shp2pgsl to directly load the shapefile into a PostGIS table, for example:
shp2pgsql -d -I shapefile.shp table | psql -U user database
If you prefer to use GeoJSON directly, there is
ST_GeomFromGeoJSON, but for this (as I understand), you'll need to parse and insert individual features from the GeoJSON file. See this answer for additional details.
You can always keep the GeoJSON file on the server for transfer to the client, or export individual features from PostGIS as GeoJSON, using
If going directly from shape to PostGIS doesn't fit your use case, perhaps include some additional information.