In Greek practice, we usually have the Sacrament of Holy Unction on Christmas Eve with an All Night Vigil for the Feast. I'm not sure what the Antiochians do. I'll let other's answer that.
For Antiochians, it varies from parish to parish. In my parish, we serve the Royal Hours on Christmas Eve, one right after the hour, not the 1st hour at 8:00, the 3rd at 10:00, etc. The Typikon (when Nativity falls on Tuesday-SAturday) calls for Vespers to be served, followed immediately by the Liturgy of St. Basil the Great. Then the Vigil follows with Great Compline, then Matins, then the Liturgy of St. John Chrysostom. However, we do not celebrate Vespers, but we celebrate Matins around 7:00 pm and then begin the Liturgy afterwards. I would prefer that we follow the directions in the Typikon, but we would have hardly anyone there.
George, do the Greeks not celebrate the Royal Hours on Christmas Eve?