Unfortunately, there don't appear to be any airlines that fly nonstop from New York to Ecuador -- the best you can hope for is a flight with one short connection (usually an hour or so), making the journey time around eight hours. The four main airlines you'll be looking at are COPA Panama, Avianca Aerovias, TACA, and LACSA.
COPA Panama flies from JFK to Quito, Ecuador, with a brief hour-long stop in Panama City -- the total flying time is 8 hours, 6 minutes. An even better option is Avianca Aerovias' flight via Bogota, Colombia -- the layover is only 35 minutes, putting the total flying time at around 7 hours, 35 minutes. American Airlines also flies out of La Guardia, connecting in Miami, but the layover is usually a little longer, putting the journey at around 10 hours.
Bottom line: look for a flight that has one very short connection, and you'll be able to make the trip from airport to airport -- assuming, of course, that there are no delays -- in around seven and a half to eight hours. Be sure, however, that you're comfortable with having a relatively short layover time (will you have to change terminals, for example? Are you prepared to miss your plane if your preceding flight is late? Sometimes giving yourself the extra cushion is worth it, even if it means your travel time is longer.)
Hope this helps, good luck, and have a safe flight!