You probably want to use the gray dropper in the levels adjustment dialog to do white balance, as the black and white droppers are used to set the black and white points for the image. Also, why do you make a duplicate of your image (the background) prior to creating a levels adjustment layer? Do you do sharpening or something on this duplicate of the background? If you are trying to use the white point tool to set your white balance and you are clicking on a spot that already has a RGB value of 255, 255, 255, then nothing should happen - that spot is technically neutral at pure clipped white, but in reality it contains no color information to assist in the adjustment even if that pixel, in the scene, has a color cast acting on it. You need to find a spot in the image that is about midtonish in terms of luminance and should be neutral in terms of its actual color content. In general, I would not use a blown white or plugged black to set WB as there may be no actual useful color information contained in those areas of the image. Use the gray dropper, find a midtonish area of the image that should be neutral and click on that area to set WB. Also, check your "auto options" under the levels adjustment fly-out menu in the upper right corner of the adjustment layers palette. There may be differences in how your Mac and PC auto options are configured that are yielding different results for the particular approach you are using.
Here is a fun exercise to try as well, especially if you are having a hard time finding a known neutral area in the image. Since you duplicate the background, use that duplicate layer as a dummy WB layer - here's how. With the duplicate layer selected, go to Filter>Blur>Average, to average the entire image into a single color. Then create a levels adjustment layer above this and click on the average layer with the gray dropper levels adjustment tool. This should balance the average layer to neutral. Then unclick the "eye" icon next to the average layer to hide it - now the WB levels adjustment affects the background (the original image). You have just essentially performed an auto WB on the image - YMMV. Compare the results to an "Auto Tone" or "Auto Color" operation to see which fits best for the image.
Another thing to consider - if you select the dropper tool in the toolbar (not the layer adjustment dropper, the actual color picker dropper positioned directly below the crop tool) check your "sample size" and "sample" settings - choose a 5x5 to 11x11 or so sample size to start with, so you are not sampling a single noisy pixel when making these adjustments.