^I’m not sure schools can think you’re asian (or any other race) based on your first or last name. At my high school, we have many names that by the name itself appear one race, but the person is actually not at all that race.
To answer the OP’s question, I would put how you best identify. If you were born in China/visit China frequently/etc., then put white and asian. If not, it is probably in your best interest to just check white.