Historically, race has been constructed as a social divider. This was true at the foundation of whiteness and it is true today. These can be seen in historical examples throughout history, as wealthy whites, at the start of slavery, sought a solution that would separate the interests of poor whites and blacks, and this strategy was employed once more when unionization first began as working class whites were once again pitted against blacks. Race has been constructed as a difference that stands between what would normally be a more united working/middle class, but instead they are divided by a trivial difference such as race. (I say trivial in the sense that it is trivial in theory not in reality.) Considering that the greatest benefits of this racial state go to a small portion of wealthy whites I believe the argument to deconstruct whiteness would have significant benefits for many working class whites.
In this sense, the deconstruction of whiteness among working class whites would prove to remove race a social divider among most of white America. Not only this, but without such a stark division by race and a more united working class, societal changes could be made not on race but instead on class.
I think Obama is a good example of this as much of white America moved beyond an issue of race and voted for their personal interest. This same example can be applied to structures of power all over the country.
Of course this surrender of white privilege does come at its cost. There is no longer that racial underclass that working class whites would always be above. But what will happen, as a result of societal changes that are sure to come from a more united working class, is an increased possibility for social mobility, for both whites and non-whites.
The sell for upper-class whites is much harder. They will face increased competition from less class stratified society that would result from a more socially mobile middle/lower class. But they still will be upper class, which means they will still have money and as much a chance as anyone. It is capitalism after all so just compete.