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
Quite different than the NIV translation
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.
Jehovah's Witnesses use the standard 66-book Protestant Bible, but usually use their own translation thereof (they do reference other translations from time to time, but generally use The New World Translation). It's fair to say that the NWT is quite, let's say, distinctive in places, and has received a fair amount of criticism. The Witnesses do not in any way claim that the translation of the NWT was inspired by God, and are happy to argue their doctrines from other translations if you ask them to. (Indeed, they did so for many years before the release of the NWT, and continue to do so in languages which do not yet have a version of the NWT.)
Here are some of the distinctive features of the New World Translation.
In general, the footnotes and appendices in NWT Ref are about the mechanics of translation, not theology. They are about tricky linguistic points and textual variants.
Re John 1:1 According to Colwell's deirect law of Greek grammer when the predicate nominative is preceded by an adverb there is no indefinite article (ie "the"). It is on this basis that John 1:1 is trans. "Word was god" however Colwell's law is a direct law, not an absolute law,that is why in John 18:37 when we see an identical construct it is universally translated "a king" twice. The scripture, then, can legitimately be translated "a god". The viewpoint of the WTBS is that "a god" is in agreement with other scriptures that emphasize Christ's subservient role to his Father.(Compare John 20:17, Rev. 3:12)