Either a car service or a yellow cab should have no problem making a stop, as long as you tell them up front about it and are willing to pay for it (taxis charge their metered rate of 50c/min for wait time; car service charges for stops/wait time vary).

Note that if the stop is outside Manhattan, the taxi fare will be metered the whole way (and the JFK-Manhattan flat rate won't apply) even if your hotel is in Manhattan. For this reason, you must tell the driver about the stop as soon as you get in the cab so (s)he doesn't set the meter to the JFK-Manhattan flat fare. If you encountered resistance to stopping from drivers in the past, it's possible it was because the driver had already started driving under the flat fare and didn't have an easy way to switch to the proper metering.

If the stop is in Manhattan, the flat JFK-Manhattan rate will apply from JFK to the stop, and then the regular meter will start running as soon as you get to the stop. See http://www.nyc.gov/html/tlc/html/pas...cab_rate.shtml

Edited to add: Note that if this is a stop where you're the only passenger and you're going to get out of the taxi and go into a building for 10 minutes while it waits outside, then I can see how some drivers might not be comfortable with that (since they risk you disappearing and skipping on the fare). For that kind of stop, a car service where they have your credit card on file might indeed be a better option.