Hemingway, the author of this short story, contributed the modern American hero to American Literature.  One of the key characteristics of this hero is that he is thoroughly disillusioned with life, especially with the American dream, partly what makes him an anti-hero.

Krebs does not want to be a hero, yet he is the main character in the story.  He obviously fought bravely in WWI, but his family cannot get past the fact that he has changed and will never be the innocent, enthusiastic son that they sent off to war.  Krebs doesn't want responsibility, complicated relationships, or parental pressure; he cannot even summon emotion for his mother.  This disillusionment with life and his family places him in the anti-hero category.

In regards to his connection to today's protagonists, Krebs does seem similar to teen heroes who endure angst, rebel against authorities, and maintain a cynical view of society and America.  

How is Krebs an example of an antihero in "Soldier's Home"?

The term antihero is often used in terms of very unique type of figure. Typically the antihero is a character who lacks heroic qualities, but they might also be characters who bear all of the qualities of a traditional hero but are oriented toward an unworthy goal. Christopher Marlowe popularized the second type in his tragedies. Tamburlaine and Faustus, for instance, have tremendous ambition; they throw themselves into their endeavors wholeheartedly, and they stand apart from others. The antihero will stay committed even to the end and speak with a force that displays an intellectual and rhetorical superiority. The same is true of Milton's Satan, who offers an amazing display of epic heroic qualities in the first two books of Paradise Lost. In the Romantic period, Byron's Don Juan might fill that same role. These are appealing protagonists, but they are not exactly heroes in that they do not embrace the cultural values deemed appropriate.

Krebs, like many Modern protagonists, lacks the energy and strength of personality to match Marlowe's and Milton's antiheroes. His listlessness and lack of direction complicate his connection to the archetype. However, in the Modern tradition, the antihero will often appear as someone like Krebs, who has seen what society's embrace of war and ideals of masculinity offer and who has decided to reject them. He is willing to hurt his mother's feelings and to disengage from his former friends, increasingly isolating himself from the conventional world.

Throughout the story, the motif of not fitting in marks Krebs. He has come home from the war too late to experience the celebrations given to soldiers, and he has no interest in pursuing post-war America's quest for status and wealth and domestic life. Krebs has no capacity for post-War earnestness. In all these ways, he embodies a visual rejection of traditional heroism. In this way, he seems to have more in common with a Hamlet or Prufrock figure; Hamlet himself declares, "No, I am not Prince Hamlet nor was meant to be." Krebs similarly rejects these simplistic identities. Sadly for Krebs, as is again true of many Modern protagonists, it is easier to resist and reject than to find something to which one can be passionately committed, even if that commitment is foolish or tragic.

How is Krebs an example of an antihero in "Soldier's Home"?

Krebs is an antihero because he does not have the traditional heroic qualities such as self-sacrifice, charisma, and purpose. He is selfish, listless, reserved, and lacking direction in life. He is a member of the "Lost Generation"--the generation that was haunted by World War I. Krebs seems lost in his own thoughts, and spends his days looking at pretty girls, going to the pool hall, and reading about the battles he was in. He is not sure of his role in the war, and he is not sure about his role in his family. He does not fit in.  Yes, he did demonstrate sacrifice by serving in the army, but after he comes home he seems selfish, and he says that he doesn't love anyone, not even his mother. Krebs is not the good-looking, all-American hero that gets the girls; he is disillusioned about life and uncertain of his future.

How is Krebs an example of anti-hero in "Soldier's Home"?

Unfortunately I had to edit your question as it actually asked multiple questions. Enotes regulations only allow you to ask one question, so I have focussed on your first question.

When we consider the term "anti-hero" we are thinking of a protagonist of a text who appears in much modern literature. The anti-hero contrasts sharply with the traditional figure of a hero, who responds to challenges and conflicts with courage and self-sacrifice. The anti-hero however surrenders to forces such as inertia, hopelessness and disillusionment. It is clear that when we examine the character of Krebs with this definition in mind, he fits the role of an anti-hero. He is disillusioned, passive and defeated by life. Note the concern of his mother and how he is compared to Charley Simmons, another young man of his age:

"Your father is worried, too," his mother went on. "He thinks you have lost your ambition, that you haven't got a definite aim in life. Charley Simmons, who is just your age, has a good job and is going to be married. The boys are all settling down; they're all determined to get somewhere; you can see that boys like Charley Simmons are on their way to being really a credit to their community."

However it is precisely this that Krebs is unable to do. After his experiences of the horrors of war, he is unable to "settle down" and "get somewhere," because he has decided to not participate in life, finding it easier to have no commitments and to live life on the sidelines. All of which, of course, marks him out as a typical anti-hero that does not conform to society's demands and pressures.