This is "Paladin of the Lost Hour", an episode of the 80s version of the Twilight Zone, and written by Harlan Ellison (based on a novelette of his).
From Wikipedia's description, we see most of the details you remember, including a Vietnam vet meeting a strange old man:
An old man standing at a grave, apparently grieving, is suddenly attacked by a couple of muggers. The man screams that someone must protect him. One of the muggers takes the only thing the man had—a pocket watch that starts to glow and burns the hand of the mugger. It floats through the air back to the old man, while another man visiting the grounds helps him. The old man, who reveals his name is Gaspar, wants to talk to Billy, the man who helped him. They go to Billy's apartment and talk about what happened at the cemetery. He goes there to visit his "girl" and Billy was visiting a friend's grave. Billy must go to work and lets Gaspar stay so Gaspar can rest.
Billy gets home to find Gaspar still in the apartment and cooking dinner. Billy discovers that Gaspar is homeless and dying. He offers to let Gaspar stay, and Gaspar discovers that Billy was visiting the grave of a man he fought with in the Vietnam War.
Calendar reforms leading to the 'lost hour':
At the cemetery, Gaspar tells Billy the story of how a pope from the 16th century decreed that the entire civilized world would observe the calendar in line with solstices and equinoxes. But the pope's calculations were off by one hour. That hour "slipped free and bounced through eternity." An hour that must never be used. An hour that must never toll. To do so would bring eternal night, which would happen when his pocket watch turns twelve, midnight.
Gaspar tests him by asking Billy to allow one minute for Gaspar to call back Minna and enjoy one minute with her. Billy has an ethical problem with the request, and refuses it. Gaspar says that Billy's response was the correct answer, and that he passed the test.
Meeting an old soldier who died saving him:
Gaspar then says he is going to give Billy a gift and lets the watch tick. A wild wind comes up and Billy has a vision of a Marine in dress uniform. Although he says nothing, Billy smiles knowingly. As the Marine and the wind disappear, Billy notices Gaspar on the ground. He tells him that the Marine let him off the hook, relieving him of his guilt. The other man never knew he saved Billy's life, and thought he'd died in vain, but now knows otherwise. Gaspar opens his eyes and asks, "May I remember you to my old girl?" He gives Billy the watch and then dies.