Kalliniks’s name day is July 29, the feast of Kallinikos.
Churches that follow the Old (Julian) Calendar keep the same feast on August 11 of the civil calendar. See everything commemorated on July 29.
The saint behind the name

Kallinikos
July 29, Martyr, Καλλίνικος
St. Kallinikos was a Christian of Asia Minor who traveled from place to place preaching Christ and turning many from the worship of idols. Seized during the persecutions, he was brought to Gangra in Paphlagonia, where after cruel torments he was condemned to death by fire, and gave up his soul to God with thanksgiving.
Name days: Kallinikos, Kalliniks
Also celebrating on July 29
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.