Converting to GeoJson, you need the geometry, so the .shp should be sufficient plus the .prj for the projection. The other extensions provide indexing capabilities and attribute table information
Esri's definitions for .shx and .dbf is the following:
.shx—The index file that stores the index of the feature geometry; required.
.dbf—The dBASE table that stores the attribute information of features; required. There is a one-to-one relationship between geometry and attributes, which is based on record number. Attribute records in the dBASE file must be in the same order as records in the main file.
Another way to understand those extensions
.shx: ESRI and AutoCAD shape index position. This type of file is used to search forward and backwards. Mandatory.
.dbf: Standard database file used to store attribute data and object IDs. .dbf can be opened in Microsoft Access or Excel. Mandatory.