Fay is a private school for grades K to 9 that has a boarding option in 7-9th grade. Some of the boarding students are international.
If not JBS, look for feeder schools to day/BS you are interested in. Case in point. Pike School is a day school in Andover, MA. It is also a feeder school for PA, meaning many kids who start KG at Pike end up at PA. They take quite a few kids from Pike’s 8th and 9th grades.
If you want a school like Milton Academy, start them out in KG or primary grades where there are more spots availale. Be willing to foot the bill for private school to get them in early. We had a plan to send our kids to BS in 9th, but by the time they applied, there was only something like 1 male and 1 female spot available in 9th grade at MA. Look at the schools and if you want a private day school, start 'em early when more places are available.