No, Jack White didn't have a sex change, but he is now rapping.
The picture of the women is artwork for Jack's single "I'm Shakin'." On the B-side of that single you'll find a song titled "Blues on Two Trees," and Jack is rapping. Here's a 30-second sample.
Here's the video for "I'm Shakin'" you can find this on his latest album Blunderbuss.