How many hours is the dog crated at night? A dog that poops while being led out the door is desparate to go... I assume when you go to her crate to let her out you immediately put a lead on her and hurry her out the door.. no stopping.
When she poops in the appropriate place do you reward her with food and praise and treat her like she just produced a pile of solid gold? That is very important. You need to be present EVERY TIME she poops and feed her several small treats and praise her.. and don't worry.. the neighbors already KNOW you are crazy..
When do you walk her? I suggest you feed her in the morning AFTER she has pooped and again in the afternoon no later than 5:00PM. Take her out for a good LONG walk and play with her about 1 hour after she eats. When I mean a good long walk.. I mean out for an hour... at least. Before going to bed, take her out again.. be WITH her and don't forget the copious happy high voice kind of praise when she goes (this goes for pee too, FWIW).
I also suggest you get up in the morning and walk her at least 1/2 hour.. and yup.. in the dark with a flash light. I do it every day (nights too). It is highly inconvenient but dogs are not always convenient. LOL
Be sure to have a set schedule for this dog.. both the feeding and the poop/pee walks. You need to be punctual and your neighbors need to be able to set their watches by when you walk your dog.
Does anyone interact with her during the day? Is she brought in at all during the day? Who plays with her during the day? A dog outside all day in MA in the winter in a yard is not acceptable. It is too cold this time of year, unless she has access to a heated indoor area.
When she is in during the evening when you are home, she is inside with you, right? You play with her and give her lots of attention? I only say this because it never ceases to amaze me the number of people who crate their dog every minute it is indoors. I am not saying you do.. just mentioning it in case you do.
I would say for this dog to soil her crate she is telling you something and that something is she is REALLY desparate to go.