What triggers the boy's fear of death in Hemingway's "A Day's Wait"?

Little Schatz begins feeling sick early one morning and the illness spawns a giant misunderstanding that leaves him waiting to die in Ernest Hemingway's short story, "A Day's Wait."

    He came into the room to shut the windows while we were still in bed and I saw he looked ill. He was shivering, his face was white, and he walked slowly as though it ached to move.

The father, the narrator, sends his son back to bed.

But when I came downstairs he was dressed, sitting by the fire, looking a very sick and miserable boy of nine years. When I put my hand on his forehead I knew he had a fever.

A doctor is summoned, and it is determined that Schatz has a case of the flu. His temperature is noted and medicine is prescribed. But it soon becomes evident that Schatz has something else on his mind.

    'About what time do you think I'm going to die?' he asked.
    'What?'
    'About how long will it be before I die?'

Schatz has been worried about dying from the flu (which is a slight possibility, of course), but he has been assured that he will soon be better.

    'You aren't going to die. What's the matter with you?'
    Oh, yes, I am. I heard him say a hundred and two.'
    'People don't die with a fever of one hundred and two. That's a silly way to talk.'
    'I know they do. At school in France the boys told me you can't live with forty-four degrees. I've got a hundred and two.'
    He had been waiting to die all day, ever since nine o'clock in the morning.
    'You poor Schatz,' I said. 'Poor old Schatz. It's like miles and kilometers. You aren't going to die. That's a different thermometer. On that thermometer thirty-seven is normal. On this kind it's ninety-eight.'

Schatz had overheard some French friends referring to the inability to live with a 44 degrees temperature. What he did not understand was that they were using the Centigrade scale, rather than the American Fahrenheit calculation.

What triggers the boy's fear of death in "A Day's Wait"?

The overhearing of the conversation between father and doctor is what triggers the boy's concern over dying.  It becomes clear that the boy perceives the conversation between the father and the doctor in a fatal light.  When he hears the disclosure of a 102 degree temperature, the boy confuses this with his own thoughts about the capacity for human temperature.  The immediacy of death becomes real at the convergence with the boy's own misunderstanding of the difference between Fahrenheit and Celsius readings of temperature as well as his father leaving him.  Driven by both realities, the boy becomes concerned over his death.  

The boy believes he is going to die and that the "wait" is for death.  The boy becomes overcome with a reality over which he has no control, but something that obsesses him nevertheless.  It is in this where the ending of  how "he cried very easily at little things that were of no importance" becomes relevant.  The dwelling on death over misunderstanding and elements that fundamentally represent "no importance" is what causes the boy to become concerned over dying from the very exposition of the story.