Eustace’s name day is September 20, the feast of Eustace the Great Martyr.
Churches that follow the Old (Julian) Calendar keep the same feast on October 3 of the civil calendar. See everything commemorated on September 20.
The saint behind the name

Eustace the Great Martyr
September 20, Martyr, Ευστάθιος ο Μεγαλομάρτυς
St. Eustathios, called Placidas before his baptism, was a Roman general of the second century. According to tradition he was converted while hunting, beholding a vision of the Cross between the antlers of a stag; he was baptized with his wife Theopiste and their two sons, endured the loss of all his possessions and family, and after years of trial was reunited with them.
Name days: Efstathios, Efstathis, Stathis, Eustace, Eustathius, Efstathia, Stathoula, Theopisti, Theopiste
Also celebrating on September 20
These names share the feast, many of them other forms of the same name:
Efstathios, Efstathis, Stathis, Eustathius, Efstathia, Stathoula, Theopisti, Theopiste
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.