Julian calendar is obsolete and in present time no one uses it. Julian calendar was replaced by Gregorian calendar.
According to Gregorian calendar Guru Gobind was born on 1st January 1667. Either we follow Julian or Gregorian calendar the Hindu date of birth of Guru Gobind falls on the same day. According to Hindu calendar it was Saptami, Paush, Shukla Paksha, 1723 Vikram Samvat when Guru Gobind Singh was born. The date of birth of Guru Gobind Singh has no controversy which is often found with other Gurus and Saints.
Nanakshahi calendar which was developed recently fixed the Guru Gobind Singh's Jayanti on 6th January and later revised it to 5th January. This date has sparked on-going controversy and it seems that currently Hindu calendar is getting used to fix the Jayanti of Guru Gobind Singh similar to Jayanti of Guru Nanak which is always fixed on Kartik Purnima according to Hindu calendar.