I found the title of that beautiful, eerie and haunting song that plays at the start of Winter's Bone movie. It's called Missouri Waltz. This version is A Capella and sung by Marideth Sisco. Now the setting and the interpret are important to the crazy atmosphere, but the song is pretty haunting in nature. Here's an interpretation made accompanied with acoustic guitar, sung by the timeless voice of Johnny Cash. It's not as eerie, but it has very strong character.
Hope you're having a great Saturday everyone!