on the type of snare, as AZ Dawn says.
If a rabbit squeals it sounds like a child in terror. It is a hair raising sound and spooky if you've never heard it before.
Like you say, it depends. When my friends and I snared rabbits, we were little boys and sometimes didn't get it right. Sometimes the snare wouldn't close until the rabit was almost through the loop so it would hang there by one paw. And yes, then it would scream until one of us nailed it in the head with a stick. Of course since the winters were very cold (-40), any rabbit that didn't die quickly was frozen stiff within 20 minutes.
I know that sounds cruel but bear in mind this was 50 years ago in a small northern community half of whom were Cree or Metis. You would not believe how proud a little boy of 9 could be and what a fuss his family made over him when he brought home a couple of rabbits.
If you liked a particular girl, you would cut off one the paws and give it to her. Thinking back it was a sort of cross cultural message, the rabbit's paw was good luck from the Irish/Scottish perspective and from the Cree/Metis perspective it meant you would be a good provider.