From Pediatrics:
Breastfeeding was inversely associated with reduced risk of neonatal respiratory tract infections in girls but not in boys. Breastfeeding may confer protection against some community-acquired infections as early as the first month of life.
The authors can't come up with a mechanism for why there is this difference - but they suggest that these results be interpreted with some caution. The bottom line is that breasfeeding is good:
Currently, exclusive breastfeeding is recommended for approximately the first 6 months of life and breastfeeding with complementary solid foods for at least the first year of life.28 However, in the United States, most mothers do not breastfeed for this duration.29 In counseling mothers of newborns about their infant feeding choices, our data suggest that the protective effects of breastfeeding start during the first month of life and that even a short period of exclusive breastfeeding may benefit young children.