If the goal of AA is to increase the percentage of minorities in law school and therefore these minorities have some advantage (and this gain to the minority is a loss to the majority), then by admitting to being white, you hurt yourself.
The target (number published) is a percentage of minorities. If you mark “I prefer not to respond,” then by admitting you, the school does not decrease its minority enrollment, because only applicants who reveal their race are considered in this percentage. So by doing this, you give yourself an advantage over self-described white people, that is, you become race-neutral. Being race-neutral means you only affect the other thing besides diversity that law schools care about, numbers.
I obviously do not have any hard data to back this up, but I did not reveal my race on any apps and have done better than I thought I would have.