I have been dating "Jill" for 2 years, and whenever we have an argument, and she is wrong, then her way to win is to say, "Well, I think that we should break up." I reply, "I can't control you, and you have to make your own decisions." But, after a couple of days, she comes back, says she's sorry, and then acts as though nothing has happened. How can I effectively neutralize this? Do you think her only goal is to hurt someone's feelings? What should I do? It's so disruptive that I wonder if I should go and find someone else! We are both 40, divorced, and have two children each.
You and Jill have a style of arguing that is unhealthy. In any relationship, both parties need to take responsibility for half of the problem. That's 50/50, not 51/49, etc. When you do this, the issue of blame becomes irrelevant. Getting past the blame is the first step toward listening.
Our guess is that you have to have a "winner" or a "loser" in the argument. However, an argument occurs because of perceived injustices. There is no winner or loser. The person who flees is usually left with no option but to flee because they are not being listened to. Your response to her wanting to break up is to further throw salt into the wound by acting as if you don't really care.
If you really care about Jill, and Jill about you, then you both need to learn how to have constructive arguments. Joint counseling would be very beneficial to help you learn to communicate.