I think the boost is very marginal. If one can get into H/Y/P

from these boarding schools, s/he most probably will get there from a solid public high school as well.  In addition, there are much more legacy admissions in a boarding school that you don't even know until you know.  If less competitive schools such as ND/Emory/Vandy/Georgetown/CMU (non-CS) are concerned, yes the boarding schools may be more helpful, but do you have to do the boarding to get there?  Much broader and longer perspectives should be considered in the process--not just colleges!