Like the No Name Woman, the quiet girl in Kingston's Chinese class is not given a proper name. You remember, the girl who Kingston more or less beats up in the kitchen? This girl is significant because we see how Kingston's insecurities about being a Chinese girl surface. The things that Kingston claims to hate about this girl are also the things she fears are true of herself. This is why this quiet girl is a foil for Kingston's character. (See the section on "Character Roles" for more on this.)
Kingston can't stand that the girl is so quiet. Kingston can't stand the double standards that Chinese American women face, such as the stereotype that Chinese women are quiet and therefore submissive. Kingston's resentment for this stereotype and its figure in her life are all projected onto this quiet girl in the bathroom. We see that Kingston hates this person only because of what she represents. Kingston realizes all this as she's beating the girl up and ends up in tears herself. It becomes clear that the silent girl is quiet on her own terms. She is quiet even amongst friends, so maybe that's just how she likes to be.