If you're looking for a meeting place or a good stop around ½ or halfway between Biddeford and Amagansett, you can try Southbridge, MA, which is about 3 hours southwest of Biddeford, or almost 3 hours north of Amagansett.
Other possible places to stop near the midpoint include Webster, and Dudley.
About ¼ or a fourth of the way, you can stop at Lowell, MA. This is about 1½ hours southwest of Biddeford.
About ⅓ or a third of the way, you can stop at Marlborough, MA. This is about 2 hours southwest of Biddeford.
About ¾ or three fourths of the way, you can stop at New London, CT. This is a little over 4 hours southwest of Biddeford, or almost 2 hours north of Amagansett.
If you drive straight through without stopping at all, the total distance is 272 miles. The distances shown above don't include local roads to get into each town, so if you're planning a long road trip, make sure you factor in traffic and road conditions, rest stop breaks, and interesting detours along the way. If you're in a rush, you can do the trip with zero stops, and your total travel time would be around 5 hours, 56 minutes.