I am an engineer in NYC. Good public schools are Bronx Science (
The Bronx High School of Science) and Stuyvesant High School (
stuy.edu: about Stuyvesant) and High School for Mathematics, Science and Engineering at City College (
HSMSE - The High School for Math Science and Engineering). They're more geared towards solid college prep and are among the top schools in the country - and they have strict admissions policies for gifted students. Private schools... There's Horace Mann in the Bronx and Hackley in Tarrytown (Westchester). They cost an arm and a leg, possibly an ear as well.
There are a lot of engineer outreach programs for high school students. There's the Future city competition, engineer's week, etc. Do some searching around the NSPE and ASCE websites... start with this one
2010 Engineers Week Programs: February 14–20.
Good luck!