I am currently researching late 1960s America in order to being writing a story. The story is set in 1966 in New York. So were schools (specifically Middle Schools) in New York desegregated by this time?
Yes, middle schools in the State of New York were desegregated in 1966.
School desegregation was decided for the entire United States in the famous case Brown vs. Board of Education in 1954, but even prior to that segregation was forbidden in the State of New York. (For a visual of pre-1954 segregation laws by state see map here.)
Add to this the reality that more integrated schools existed in the northern states generally (see here), and Brown vs. Board of Education was largely targeted at southern states. No timeline was given in the court decision for the implementation of desegregation, but the parties returned to court in 1955, known as Brown II, to decide the implementation timeframe.
Some stubborn holdouts in the southern states resisted until 1959, with one county in Virginia going so far as to close all public schools indefinitely. Further litigation in 1964 resulted in reopening these schools - after five years of inactivity - but with the integration policy enforced. (My point is that these are the most extreme cases, far from New York, and still they were integrated before 1966.)
All that being said, there was - and still is - an uncomfortable amount of racism among (some) students in the United States public schools. And the intolerance divides not only black and white, but also hispanic, asian, and native american... New York City has always been a blend of races trying to work together. (Believe it or not, in the northeast states there were white people who were racist against other white people!) And racism cuts both ways, so don't forget that in your story.