I'm getting a lot of mixed information without any significant sources.
According to Wikipedia he was born in Kibbutz Degania Alef, but this claim isn't sourced.
Meanwhile, Google gives me several examples (like this one) claiming he studied and was born in Bulgaria.
Considering the importance of Dayan in Israeli history, I find it strange for there to be such a lack of sources on either side (or at least, I can't find any). So, can anyone settle this question definitively?
Israeli historian Benny Morris writes that Dayan was born in Degania Alef in Righteous Victims.
Mordechai Bar-On, who wrote a biography of Dayan called Moshe Dayan: Israel's Controversial Hero, also writes of Dayan's birth in Degania Alef. Dayan's parents were from Ukraine, not from Bulgaria.
The Bulgaria myth is a common one, which apparently originates with some kind of chainmail about Bulgarian actions during the Holocaust that ties up the story by saying Dayan came to Israel from Bulgaria, as some kind of miracle for his survival. It typically relies on a book it attributes to "Michael Bar Oar" called Beyond Hitler's Grasp, which was actually written by Michael Bar Zohar, but Zohar does not claim that Dayan was born in Bulgaria at all. The rewriting of the history on the subject appears to be a rumor started with the goal of rewriting Bulgarian Holocaust history, and has confused a number of well-meaning people, but not real historians. It is definitive: Dayan was born in Israel, in Degania Alef.