[David L.]

1. Do they have a hospital there? Is there a hospital on the island?

2. Do they have any hospitals there? Are there any hospitals on the island?

(We try to avoid repetition of words as in 1. Is there any hospital there?)

If a native speaker said, "Is there any hospital there?" it would be for a specific reason/effect. He would be saying, Is there anything like a hospital in that godforsaken place/anything that passes for a hospital in....
In asking the question, he probably has a picture of this place as being so primitive, that what we regard as 'a hospital' couldn't possibly exist there. The question is asking, just how primitive are things there in the way of getting medical help? The tone of voice would be one of great doubt, as if he is saying, there can't possibly a hospital in such a godforsaken place!