The way I see it, Whites shouldn't be able to say "I'm proud of being white" but nobody at ALL should be able to say they're proud of your race. your race says NOTHING about who you are, your CULTURE does. You can say "I'm proud to be born and raised here/raised here even if not born here/ raised by people of this culture" but saying "I'm proud I'm black/white/asian/mexican(without ever living in Mexico)" etc. just makes you sound bigoted and ignorant.
ADDENDUM: Pride in and of itself is something I don't believe is necessary or a positive aspect of humanity or society, but it is naturally unavoidable, so instead I list to what extent one should express pride, at least in my own opinion.