Well, Jesus birthday aside, the Maccabee War occurred nearly 200 years before Jesus birth so any correlation doesn't seem to add up. Hanukah really is it's own celebration not at all related to Jesus.
Ok, I'm finally getting back to this. Yes, the backstory of Hanukkah has to do with the Maccabee revolt, but that doesn't mean it doesn't have any prophetic meaning. God can use any physical event to point toward a future spiritual truth, and the entire bible is filled with examples of foreshadowing and prophetic events.
God is able to orchestrate all sorts of things, because God is sovereign over everything.
It's no coincidence that Jesus was very likely conceived during the time of Hanukkah, because that means it has a deeper spiritual significance. It was the darkest time of the year, and Jesus is called the Light of the World, and Hanukkah is also called the Festival of Lights.
And the only time Hanukkah is mentioned in the entire bible (as far as I've seen) is actually in the New Testament, when Jesus Himself observes Hanukkah. Not only that, but at that time was when He made a very strong statement that He is the Messiah. (John 10:22-30)
There's actually much more to be said about this, but I think others can do a better job explaining it than I can. So I'll try to find a video or article about the connection between Jesus (Yeshua) and Hanukkah.
Hanukkah is literally translated into "dedication" and it is referring to the uprising of the Jews against the Syrian-Greeks who had at one time allowed Jews to practice their religion but were now trying to clamp down, Hellenists trying to have them worship false G-ds and engage in general idiolatry, a serious in in Judaism.
This war was fought and won by the Jewish Freedom Fighters referred to as the Maccabees. Often endeared with the phrase "the light overcoming the darkness". They restored the temple which the pagans had destroyed and continued their practices without fear of being killed for their beliefs.
Some interesting debates, of which there are countless in the Talmud and within the Jewish community; apparently, according to a well known Rabbi in Jerusalem who I listen to often, the oil for the menorah was only supposed to last for one day, but ended up lasting for eight. Hence the eight days of lighting, though he suggests that in historical books on this period, it was not a literal event but a symbolic based on the details so the oil story doesn't seem to be a literal, but that's just his statement based on his readings.
Also, according to these historical documents/books which dissect this historical period, pieced together, the war continued for 25 years though it isn't always taken to be that length by the modern follower. I have read from different sources that it lasted from 7-40 years. This is the challenge with history which was often spread by word of mouth in that time.
I'm not sure about the relevance to Jesus who wouldn't be born for nearly two centuries if my memory is correct. Neither the two should overlap though I suppose you could associate one with the other. This isn't new though is it? This is why it is safe to refer to the period as the "Holidays".
Yes, as I said, it doesn't matter that the physical aspect of it was an event that occurred before the time of Jesus, because the whole point is that it has prophetic significance. Do you disagree that God is able to orchestrate things in advance?
Anyway, not all Christians agree with my view that Jesus was conceived in late December, during the time of Hanukkah / Christmas. I'm not claiming that this is a standard Christian view. But a lot of times, one has to dig deeper, in order to see all the cool foreshadowings in the bible and physical events that has prophetic meaning. I hope that made sense to you.
because that's what I posted in reference and source.
Merry Christmas to you too (it's still Christmas here in the US, even though I think it's already tomorrow where you are) and Happy Hanukkah! 
