Do Jehovah's Witnesses have their own version of the Bible? Do Jehovah's Witnesses have their own version of the Bible? What is their reason for not accepting other versions such as the NIV or the KJV?
Yes, they do. It's called  New World Translation.

There are several verses with a completely different meaning than other translations, such as John 1:1


  1 In [the] beginning the Word was, and the Word was with God, and the
  Word was a god


Quite different than the NIV translation


  1 In the beginning was the Word, and the Word was with God, and the
  Word was God.


The difference is easy to see when you consider how it affects the doctrine of the Trinity, Christ's identity, and the idea of a single God vs. the idea that the word was "a god" - separate from The God.

Other, commonly accepted translations agree with the NIV translation.

The argument I've heard time and time again is that the Bible has been corrupted over time, intentionally, or not, and that they, the one true Church, have the only accurately translated version. (This was from three personal friends, all JW, and from several missionaries that I spoke with. I don't have official statements from the Watchtower Society claiming this.)

There are more differences, along the same vein, but I don't wish to get into a discussion on the validity of their claims vs. those of other denominations. Such discussion would be off-topic and frowned upon. 

I included the above, only to show that the NWT is more than a slightly differently translated version. Most of the passages are the same,but those few that are different significantly alter the meaning relating to some foundational doctrinal views held by more (traditional? mainstream?) denominations.