Gosh I had no idea there was actually a term for this "singing" I have some LOUD ladies and I get paranoid about it and like you run out and thrown them scratch and goodies. (That is why they are probably singing!!) When mine are the loudest however is after one just laid an egg and she is separated from the other girls, she will stand there are create such a racket until the other 3 come running. Sometimes it is so loud that I am out there trying to help her find the others. I have gone over the deep end!! Out there helping chickens find their friends...... as my BIL says to my sister who also has a flock, " I wish I was a chicken and get all this attention, goodies and nice little chats!!"
Good Luck with the neighbors, have you brought them over fresh eggs?? Maybe that is just the thing to pacify the whole song thing??