As soon as he introduces Schatz, Hemingway says,

He was shivering, his face was white, and he walked slowly as though it ached to move.

Despite his repeated laconic assurances that he's okay, Schatz seems very detached from what is going on around him and cannot follow what his father reads. He looks strangely at the foot of the bed and refuses to try to sleep.

Schatz sends his father away and shuts himself up in his room, refusing to let anyone enter. He tells his father not to come in, because "You mustn't get what I have." When his father does see him, he is still white-faced, but the tops of his cheeks are flushed by fever. He is still staring at the foot of the bed in the same way as before. Hemingway continually emphasizes this fixity of gaze, suggesting that the boy is trying very hard to retain control over himself:

"I’m taking it easy," he said and looked straight ahead. He was evidently holding tight onto himself about something.

Finally, just before the end of the story, he asks the question that has obviously been preoccupying him all along:

"About what time do you think I’m going to die?" he asked.

All the comments on his pallor, his distraction, and his fixity of gaze, as well as his refusal to let others enter his room, contribute to the tension which rises to this point and is released when his father explains his mistake.

Which of the boy’s words and actions give clues that he believes something is terribly wrong?

First, when the boy indicates that he'd "rather stay awake" than go to sleep when he is very clearly quite ill, this seems to provide a clue that he believes something is wrong. Otherwise, why not go to sleep and be more comfortable? He, perhaps, fears that something bad will happen if he falls asleep.

Then, the boy tells his father, "you don't have to stay if it's going to bother you." We might wonder what the "it" is; the boy seems to anticipate that his father will be bothered by something that has not yet occurred, and he is obviously not talking about getting better. He must fear the idea that he is going to worsen and perhaps die, and his death would be the "it" from his thoughtful statement.

Later, the boy will not let people into the room with him, saying, "You mustn't get what I have." This concern for others' health and safety is another clue that he believes his illness to be very bad indeed.