No, there is NO requirement that a EGPWS unit have any sort of GPS. Quoting from the report I linked to:
The certification of the TAWS aircraft installation was guided by the JAA document “Section One: General Part 3: Temporary Guidance Leaflets LEAFLET NO 12: Certification Considerations for the Terrain Awareness Warning System: TAWS.” – referred to as TGL12. This allowed the use of the aircraft navigation system, which was designed for area navigation, to be the source of positional information for the TAWS. The capabilities of such area navigation systems vary from very poor to very good. There are no minimum positional accuracy requirements imposed on the source of positional information for TAWS, other than by reference to relatively relaxed area navigation requirements, and no requirements to supply TAWS with relevant indicators of navigational data quality. There is no requirement to have GPS as a source of positional information for the TAWS. The standard GPWS alerts, pre-TAWS, are viewed as fall back protective modes in the event of a failure of the new TAWS database based modes.