What does Regis mean?
From Latin 'regis', the genitive of 'rex' — literally 'of the king', i.e. royal/kingly. Popularized partly via Saint Jean-Francois Regis. Means 'of/belonging to the king' rather than 'king' itself.
What does Ignatius mean?
Roman/late-Latin name popularly associated with Latin 'ignis' ('fire'), giving the sense 'fiery, ardent'. The link is disputed: the name may actually derive from the Etruscan family name Egnatius, with the fire meaning being a long-standing folk etymology.