Speaking as an American: People sometimes say, "You should visit such-and-such a place. It's beautiful there. It's God's country." It almost always refers to natural beautyIt almost always refers to natural beauty. To the extent that a person means it literally, he means, "this is a place that particularly showcases God's creative power".
BTW I don't think I've ever heard it said as "God's own country", but always simply "God's country".
While many Americans believe that America has been particularly blessed by God, we don't say "America is God's country" or "God's own country" to express this idea. I'm rather surprised by the definition and examples you quote, because I don't think I've ever heard the phrase used that way. Of course I can't say that no one, anywhere, ever said that. But it's not common usage today. And I've read plenty of old books and I've never noticed it being common usage from the past. And as a right-wing Fundamentalist who routinely associates with others of like mind, you'd think that if anyone was using these words this way, it would be me and my friends!
I'm a little suspicious of those example sentences, especially the first one, "such a situation in the God's own country". No fluent English speaker would say "the God's". It would just be "God's". "God" with a capital "G" is a proper noun and thus does not take an article. So where did this sentence come from? Is it something an American actually said? Or something that a non-American thinks is the sort of thing an American might say? Which, of course, is highly unreliable.
One wouldn't have to be a Jew or a Christian to use this phrase. Lots of other religions believe that there is some sort of God. I'd be surprised to hear an atheist say it, as it implies a belief in God. Atheists generally avoid phrases that refer to God, except when used as a swear word. I suppose an atheist might think of it in some literary or metaphorical sense.
I don't see why this phrase would be considered offensive. I don't see why this phrase would be considered offensive. Of course there are people who go out of their way to find things to be offended by, so I suppose an atheist could declare that he's offended by the mere mention of God. But if you're going to avoid saying anything that might even indirectly imply that you disagree with another person about any conceivable subject, you'll have a hard time speaking at all. Perhaps someone could be offended that you think that your home is somehow special to God in a way that his is not. But again, this is quite a stretch. If you simply said, "Oh, my homeland is a beautiful place with lovely trees and mountains", would he scream, "How dare you say that your homeland is beautiful! Are you saying that mine is not?" That would be pretty irrational. Not to say that people aren't irrational.