The correct phrase is “published on date”. While “publish in date” is a common error, it is not correct.

The Oxford Dictionary online gives a definition of the word “published” as being:  “To make something available to the public through the printing, TV, radio, internet etc.    ”

The word “in” only exists in phrases such as “in date” and “in time”.
The Oxford Dictionary online gives a definition of the word “in” as being:  “Expressing location, presence within a region or area.”

This implies that the word “in” is used to define location, where as “on” is used to define a point in time.