The Brigham and Women’s Hospital (Boston) has a paid Youth Summer Program for Boston students 16 and older. http://www.brighamandwomens.org/research/opportunities/documents/1.pdf
Teens 15 and older can volunteer at Mount Sinai Hospital in New York City http://www.mountsinai.org/careers/volunteer-services/how-to-become-a-volunteer
The Bronx-Westchester Area Health Education Center (New York City) has six-week unpaid summer internships for high school students interested in the health professions. http://bwahec.org/summer-health-internship-program/
So does the Manhattan-Staten Island Area Health Education Center http://msiahec.org/programs-for-high-school-students/ and the Brooklyn-Queens-Long Island Area Health Education Center http://www.bqliahec.org/ship.htm.
Children’s Hospital Colorado (Denver area) has a two-year high school internship program which includes field trips, mentoring, pre-collegiate assistance and paid research internships at the hospital. http://www.childrenscolorado.org/about/careers/high-school-internships
Kaiser Permanente’s Summer Youth Employment Program (SYEP) and LAUNCH Program place high school 11th and 12th graders and recent graduates into paid summer jobs at Kaiser hospitals. The program is primarily intended for underserved students.
Northern California: http://share.kaiserpermanente.org/article/ncal-launch/
Southern California: http://community.kp.org/be-informed/program/summer-youth-employment-program
Here’s how to volunteer at Marin General Hospital: https://www.maringeneral.org/about-us/volunteer