Iezekiil’s name day is July 23, the feast of Ezekiel the Prophet.
Churches that follow the Old (Julian) Calendar keep the same feast on August 5 of the civil calendar. See everything commemorated on July 23.
The saint behind the name

Ezekiel the Prophet
July 23, Prophet, Ιεζεκιήλ
The Holy Prophet Ezekiel was a priest of Israel carried into the Babylonian captivity in the sixth century before Christ. By the river Chebar he beheld the awesome vision of the glory of the Lord upon the chariot of the cherubim, and he prophesied to the exiles both judgment and consolation, including the vision of the dry bones brought to life, a foreshadowing of the resurrection.
Name days: Iezekiil
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.