One of the objectives of Hemingway's style of minimalist writing is to open the truth of subjects up to thoughtful contemplation. For this reason, his narrators are objective though sympathetic and distanced though astutely observant. Since Hemingway often chooses settings for their symbolic value, the narrator's observation of the surroundings reveals as much as the narration of events.
Since Hemingway aims to expose the truth of difficult situations and open them to contemplation, it may be hard to assert a "claim" in his stories, particularly since he aims to be objective and inclusively sympathetic in "Hills Like White Elephants." Having said this, the one thing that is most clear from this short story is that Jig's pregnancy is going to radically change the lives of both adults. Perhaps Hemingway's "claim" might be best describe as claiming that loose relationships that aren't bound by the expectations present in marital relationships are like time bombs waiting to go off; these kinds of relationships will shatter and destroy everything dreamed of:
[F]ields of grain and trees along the banks of the Ebro. ... beyond the river, were mountains. The shadow of a cloud moved across the field of grain ...
"And we could have all this," she said. "And we could have everything and every day we make it more impossible."
"What did you say?"
"I said we could have everything."
"We can have everything."
"No, we can't."
'Hills like white elephants' was a short story written by Hemingway in the late 1920s when US was going through a serious economic recession. Here the protagonists are given no names; the man is simply called the American and the woman is being refered to as 'Jig'. Jig, short for jigger, is an old name for a 1.5 ounce shot glass. By calling the man simply American, he is symbolizing all Americans in Hemingway's point of view. The constant reference to alcohol symbolizes the wasteful and pointless lives they were living back then; the story takes place in a bar and ends in a bar. Although they have the opportunity to enjoy the sights and nature of Spain, they decide to spend pointless nights in hotels and bars and through such references, Hemingway is criticizing how many Americans lived in vanity in the 1920s.