It is definitely possible to die of a broken heart. The body just stops trying. I think the stress of a traumatic event, such as losing a loved one, can also wear on the body physically enough for health to deteriorate, maybe past the point of no return. When people are older, especially, and they have been married to a person for over half their life-- that person might have been the only thing keeping them going. Losing that person could just be too much for their physiology to handle.
Being with your significant other for so many number of years allows both to grow so fond of one another. Doug Flutie's parents must have been together for so long. They have lived with each other, and experienced every emotion that a couple can experience in their years of being together. They have grown to love and respect each other.
Yes, I do think it is possible to die with a broken heart. I believe that after one passed away, the other knew in their heart that they would soon see each other in Heaven.
No, while it is not technically possible to die from a 'broken heart', one can die from the effects of having your heart broken. For example, one may become depressed and always stay at home, and not eat. While starvation is a long stretch, this is just an example of what I mean.
CFL quarterback Doug Flutie recently lost both of his parents, one right after the other. After Flutie's father passed away, his mother died within an hour of her husband's death. This has been equated to dying of a broken heart and yes, the science shows this can happen. When overcome with grief and an emotional trauma, it is possible for someone to "die of a broken heart".