I think that men do. There is a myth that men are not as sensitive as women. They just show it differently, meaning that they do not show it as openly because society still frowns on men who are too emotional, which has devastating consequences.
Men do not fall in love as easily as women but when they do, they fall deeply. This makes it harder for them to get over it when it goes wrong. I know multiple guys who were absolutely devastated when they could not get over the girl who broke up with them. Even if the relationship was not going perfectly throughout, they were not able to handle the separation from the woman they loved.
I think there are several reasons for this. One is that men do not have the support system that women have. When girls break up with guys, they can call their best girl friends and lament about the guy. They can list every wrong thing the guy ever did, and that makes them feel better about the breakup. However, a guy does not have the social resources to vent like this. Even if they have close friends, they do not know how to express their awful feelings, because they do not have practice communicating their feelings. The depth of their love, and the fact that they cannot express their sadness, combines to make a very miserable human being.
One way to overcome this is for guys to not ignore their friends when their relationship is going well. It’s tempting for guys, especially if they are busy, as most people are these days, to focus only on their girl. But this is unhealthy. Once the breakup happens, they have nowhere to go. And for the few friends they have, they know that their friends are not going to want to hear endless details about how miserable they are. However, the more friends they have, the more likely they will be able to find one who will be willing to hear about their heartbreak.
I remember a study that showed that little boys were shown scenes of suffering, as were little girls. The little girls cried and expressed their sadness about the scene before them. However, the little boys tried to end the scene. Not because they were impervious to the suffering, and not because they were emotionally dead. But because they were so sensitive that they could not even bear to see the people suffering in the first place.