O’Connor characterizes the children as being obnoxious. What is the significance of this choice? Would the result of the story be different if she characterized them in another way?

In "A Good Man is Hard to Find" by Flannery O'Connor, the author portrays the children as obnoxious in a way that fits in with the "meanness" of rest of the characters. Lack of kindness is a large theme within the story, so much so that when The Misfit's two associates kill the entire family, the reader is left without much sadness from the event.
One of the biggest clues to O'Connor's portrayal of John Wesley and June Star comes not from the children themselves, but rather from The Misfit's speech about Jesus:

"Jesus was the only One that ever raised the dead," The Misfit continued, "and He shouldn't have done it. He thrown everything off balance. If he did what He said, then it's nothing for you to do but throw away everything and follow Him, and if He didn't, then it's nothing for you to do but enjoy the few minutes you got left the best way you can by killing somebody or burning down his house or doing some other meanness to him. No pleasure but meanness," he said and his voice had become almost a snarl.

On the surface, this is a morally ambiguous justification for murder. However, upon deeper reflection, this passage also acts as a commentary on the state of relationships and interaction in society. "No pleasure but meanness" represents nearly every character's attitude toward others throughout the story. The children are disrespectful to their grandmother; Red Sammy is distrustful of everyone and treats his wife poorly; Bailey ignores and has a short temper with his family; and the grandmother herself acts as a "better than everyone" know-it-all. In the most telling line of the story, the grandmother—after listening to Red Sammy talk about not knowing who to trust anymore—says, "[p]eople are certainly not nice like they used to be." This overarching theme of "meanness" plays out in a way that makes it hard to sympathize with any of the characters.
Early in the story, the two children respond to their grandmother's suggestion of traveling to east Tennessee (rather than Florida) with contempt. "If you don't want to go to Florida, why dontcha stay at home," quips John Wesley, while June Star responds, "She wouldn't stay at home to be queen for a day." While this shows their level of disrespect, the lack of response from their parents further adds to the "meanness" theme in the story. Though the grandmother does not act much better, by the end of the story the reader is left to wonder how differently it might have turned out had the family taken her "east Tennessee" suggestion more seriously.
When the family visits Red Sammy's diner, the children are again portrayed as obnoxious. After June Star's dancing is complemented by Red Sammy's wife, she responds by saying she "wouldn't live in a broken-down place like this for a million bucks!" Later, when Bailey initially refuses the children's request to visit a house with a secret panel, the children "yell and scream" in order to get their way. Finally, after the car accident, they "screamed in a frenzy of delight," despite their screaming baby brother and injured mother.
In short, the obnoxious characterization of John Wesley and June Star fits in with the story's theme of "meanness." They disrespect their grandmother, throw tantrums to get their way, and appear to find delight in a potentially deadly car crash. By the time The Misfit shows up, the children—as well as the rest of the family—are so insufferable that none of their deaths evoke a strong emotional response from the reader. The one exception is the baby, though one can imagine him growing up to be just as dreadful as the rest of the family.