Why do Jehovah's Witnesses not celebrate birthdays? Wikipedia states this as a fact but does not explain further. In conversations with JWs, I understood that this is based on two birthdays in the Bible:


Job's eldest son: Job was attacked by Satan on that birthday and
Herod's: John the Baptist was beheaded on his birthday


But how do these two instances become translated to a general rule for all Jehovah's Witnesses?
To begin answering your question let me first begin by saying that the Bible makes only two references to birthday celebrations:


  Gen. 40:20-22 (NWT): “Now on the third day it turned out to be Pharaoh’s birthday, and he proceeded to make a feast . . . Accordingly he returned the chief of the cupbearers to his post of cupbearer . . . But the chief of the bakers he hung up.”
  
  Matt. 14:6-10 (NWT): “When Herod’s birthday was being celebrated the daughter of Herodias danced at it and pleased Herod so much that he promised with an oath to give her whatever she asked. Then she, under her mother’s coaching, said: ‘Give me here upon a platter the head of John the Baptist.’ . . . He sent and had John beheaded in the prison.”


Our view on the matter is take note that God’s Word reports unfavorably about birthday celebrations and for this reason we reject participating in these sort of occasions.

There's no Bible law that says that we must not celebrate birthdays. However, if we think about it principle, then we can discern if celebrating is even close to something that God would disapprove of. After all, everything that is in the Bible is there for a reason. (2 Tim. 3:16, 17)

Also, if we take a look back into history, we can observe that:


  […] "the origins of birthday celebrations lie in the realm of magic and religion. The customs of offering congratulations, presenting gifts and celebrating—complete with lighted candles—in ancient times were meant to protect the birthday celebrant from the demons and to ensure his security for the coming year […] Down to the fourth century Christianity rejected the birthday celebration as a pagan custom.”
  —Schwäbische Zeitung (magazine supplement Zeit und Welt), April 3/4, 1981, p. 4.
  
  “The Greeks believed that everyone had a protective spirit or daemon who attended his birth and watched over him in life. This spirit had a mystic relation with the god on whose birthday the individual was born. The Romans also subscribed to this idea […] This notion was carried down in human belief and is reflected in the guardian angel, the fairy godmother and the patron saint […] The custom of lighted candles on the cakes started with the Greeks […] Honey cakes round as the moon and lit with tapers were placed on the temple altars of [Artemis] […] Birthday candles, in folk belief, are endowed with special magic for granting wishes […] Lighted tapers and sacrificial fires have had a special mystic significance ever since man first set up altars to his gods. The birthday candles are thus an honor and tribute to the birthday child and bring good fortune […] Birthday greetings and wishes for happiness are an intrinsic part of this holiday […] Originally the idea was rooted in magic […] Birthday greetings have power for good or ill because one is closer to the spirit world on this day.”
  —The Lore of Birthdays (New York, 1952), Ralph and Adelin Linton, pp. 8, 18-20.


Sounds familiar? Again, these references are only to show you what birthday celebrations are connected to. In sight of what you find at Deuteronomy 18:10-14, We as Jehovah's Witnesses take our stand against engaging in what the God's word speaks unfavorably of.

Does this mean that we do not celebrate anything? No.