No. You cannot continue lessons with anyone other than the assigned Juilliard precollege teacher. They’re very strict about that. If you live in Boston, studying at Juilliard precollege is inconvenient. It is a full day on Saturday, and with the drive from Boston to NYC being 4 hrs at least, it seems too much, to me. Depending upon what instrument your child plays, a better option might be Boston Youth Symphony or NEC’s precollege program, and continuing to study with the teacher in Boston.
As for extra lessons with the assigned Juilliard teacher, I doubt that would happen. The Juilliard precollege teachers are also Juilliard college and grad school teachers, have very full teaching and performing schedules. Usually, a one hour lesson a week is considered enough. The student of course needs a lot of practice time during the week to get the most out of that one hour lesson.
It was very difficult, timewise, for my kid to do Juilliard precollege, and we only live about a 2 hr drive from NYC, if no traffic. It was, however, worth it. Amazing experience.