Circa 1958, I read a short story in a Sci-Fi anthology
Perhaps "Quietus", a short story by Ross Rocklynne, first published in Astounding Science-Fiction, September 1940, which is available at the Internet Archive. If you read it in an anthology around 1958, it must have been one edition or another of Famous Science-Fiction Stories: Adventures in Time and Space, edited by Raymond J. Healy and J. Francis McComas.
in which the protagonist is (possibly) the last human on earth.
He seems to be the last human male, and is pursuing the last human female when the avian aliens arrive:
For five days Tommy followed the lead of the girl with a grim determination. He knew now that it was a woman; perhaps—indeed, very probably—the only one left alive. He had only the vaguest of ideas why he wanted her—he thought it was for human companionship, that alone. At any rate, he felt that this terrible hunger in him—he could give it no other word—would be allayed when he caught up with her.
She was fleeing him, and staying just near enough to him to make him continue the chase, he knew that with a fierce exaltation. And somehow her actions seemed right and proper. Twice he had seen her, once on the crest of a ridge, once as she swam a river. Both times she had easily outdistanced him. But by cross-hatching, he picked up her trail again—a bent twig or a weed, a footprint, the skin of a dead rabbit.
Stumbling about the ruins of a city he spots something alive – a chicken. He's about to kill the chicken when he is suddenly vaporized.
This is the part you must be misremembering if "Quietus" is your story. The story is set in the woods, not in a city. The bird was not a chicken; it was Tommy's talking crow, Blacky:
"Tommy" was what he called himself. A long time ago, he remembered, there used to be a great many people in the world—perhaps a hundred—many of whom, and particularly two people whom he had called Mom and Pop, had called him by that name. They were gone now, and the others with them. Exactly where they went, Tommy did not know. But the world had rocked one night—it was the night Tommy ran away from home, with Blacky riding on his shoulder—and when Tommy came out of the cave where he had been sleeping, all was in flames, and the city on the horizon had fallen so that it was nothing but a huge pile of dust—but in the end it had not mattered to Tommy. Of course, he was lonesome, terrified, at first, but he got over that. He continued to live, eating, drinking, sleeping, walking endlessly, and Blacky, his talking crow, was good company. Blacky was smart. He could speak every word that Tommy knew, and a good many others that he didn't. Tommy was not Blacky's first owner.
Tommy is about to kill Blacky in a rage after Blacky scares the girl away, but an avian alien intervenes to save the crow:
"It's all your fault, Blacky!" Tommy raged. He picked up a rock the size of his fist. He started to throw it, but did not. A tiny, sharp sound bit through the air. Tommy pitched forward. He did not make the slightest twitching motion to show that he had bridged the gap between life and death. He did not know that Blacky swooped down and landed on his chest; and then flung himself upward, crying, "Oh, Tommy, I could spank you!" He did not see the girl come into the clearing and stoop over him; and did not see the tears that began to gush from her eyes, or hear the sobs that racked her body. But Tark saw.
Tark wrested the weapon from Vascar with a trill of rage.
"Why did you do that?" he cried. He threw the weapon from him as far as it would go. "You've done a terrible thing, Vascar!"
Vascar looked at him in amazement. "It was only a beast, Tark," she protested. "It was trying to kill its master! Surely, you saw it. It was trying to kill the intelligent bird-creature, the last of its kind on the planet."
The scene shifts to the inside of a spaceship hovering above, a spaceship whose captain is avian, and a bird-captain certain that he/she/it has just saved the last important species on Earth.
The two aliens are of different opinions about what happened on Earth. The female believes she was right to kill the "beast" that was threatening to kill the "intelligent bird"; the male, who is driving the spaceship, has doubts.
The spaceship of the creatures from Alcon left the dead planet Earth. It darted out into space. Tark sat at the controls. The ship went faster and faster. And still faster. Fled at ever-increasing speed beyond the Solar System and into the wastes of interstellar space. And still farther, until the star that gave heat to Earth was not even visible.
Yet even this terrible velocity was not enough for Tark. Vascar looked at him strangely.
"We're not in that much of a hurry to get home, are we, Tark?"
"No," Tark said in a low, terrible voice; but still he urged the ship to greater and greater speed, though he knew it was useless. He could run away from the thing that had happened on the planet Earth; but he could never, never outrun his mind, though he passionately wished he could.
The spacefaring aliens were described early in the story:
Tark nodded. He was truly a bird, for in the evolutionary race on his planet, distant uncounted light-years, his stock had won out over the others. His wings were short, true, and in another thousand years would be too short for flight, save in a dense atmosphere; but his head was large, and his eyes, red, small, set close together, showed intelligence and a kind benevolence. He and Vascar had left Alcon, their planet, a good many years ago, but they were on their way back now. Their outward-bound trip had taken them many light-years north of the Solar System; but on the way back, they had decided to make it one of the stop-off points on their zigzag course. Probably their greatest interest in all this long cruise was in the discovery of planets—they were indeed few. And that pleasure might even be secondary to the discovery of life. To find a planet that had almost entirely died was, conversely, distressing. Their interest in the planet Earth was, because of this, a wistful one.