No, the highlights are simply blowout because the picture was taken on a phone camera. Give me a bright enough light, a piece of dark fabric and a phone camera, and I can easily make white or near-white highlights appear on a picture of said dark fabric.
Brown highlights are common on "black" fabrics, since they are often extremely dark brown (or blue), not true black. So the dark parts are definitely black.
If the blue parts of the dress were actually white, that would require a light source with a lot of blue light, as well as a camera with very misconfigured white balance. You would have to deliberately do this to mislead. It's also impossible to do this on an iPhone (which the picture was taken with).
Lastly, we know for a fact that the dress is blue and black, because the manufacturer doesn't even sell a white and gold version.