Justin’s name day is June 1, the feast of Justin the Philosopher.
Churches that follow the Old (Julian) Calendar keep the same feast on June 14 of the civil calendar. See everything commemorated on June 1.
The saint behind the name

Justin the Philosopher
June 1, Martyr, Ιουστίνος
St. Justin the Philosopher and Martyr was born in Palestine and lived in the second century. Seeking truth through the schools of the philosophers, he found it at last in Christ, and thereafter wore the philosopher's cloak in the service of the Gospel. In Rome he wrote his celebrated Apologies, defending the Christians before the emperors and describing the baptism and Eucharist of the early Church.
Also celebrating on June 1
These names share the feast, many of them other forms of the same name:
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.