I am pretty sure the USCG would have the finial say there! A vessel offering aid, may stand by, but they are not likely to try and force them.
In my experience the use of force was avoided to the n'th degree in these cases. But my experience was usually closer to the 'home country' and not in sight of the US. In situations close to land there have been cases where the 'survivors' tried to swim for land and use of force becomes necessary - illegal entry to the country... USUSALLY the folks on these boats have been days on the water with limited food, water an shade and anyone offering aide is received pretty positively. A cruise ship or cargo vessel or other vessel is 'requested' to standby in most cases, but if weather is poor may be asked to take the 'survivors' aboard with a typical reaction being JOY as they don't realize the truth yet. Only when the cutter comes into view do they connect the dots typically. BUT every case IS different. Weather and no available cutter MAY see a cruise ship taking the folks IN TO a port .... because as I said the situation is safety of life first . . .
A btw - I've "entertained" groups of survivors for as long as a week while arrangements were made for repatriation . . . the folks lived on the flight deck with some blankets, a sun awning make shift toilet facilities and two meals a day. Generally they were very grateful . . . (and under watch of armed guards)