Why would my ten year old daughter developing underarm boils?

Happens at any age: Skin abscess/boil can happen at any age and your daughter may just happened to get one there. If it is arecurrent problem, she may have problems with hair-follicles/sweatglands there...I assume she is getting hair under her arm? She is a bit young, but may have hidradenitis suppurativa (recurrent infections of her sweatglands/follicles) under skin folds areas. Consult her doc.. Good luck.

Variety of Reasons: Perhaps clogged hair follicles. Poor hygeine. Most likely is damaging the skin with shaving and developing a secondary skin infection.Also could be secondary to irritant reactions to fabrics, deoderant etc.