Hi Dotte,
Your Unique is a Davis model H. You need a used Davis full sized shuttle or Boye 9, measuring 2-1/16". Needles are "Davis Long", Boye 10, FF, or Singer 1x2. You can use MTx190 a round shank needle though the originals have a flat. Bobbins are probably the toughest to locate, if your lucky a few might come with the shuttle. Singer bobbins will work but you'd need a Singer machine to wind on as they are shorter than Davis.
Here's an owners manual close to your Unique. This is from Sear's Burdick and whoever made the manual took pictures from many different Davis models.
http://www.sil.si.edu/DigitalCollect...f/sil10-37.pdf
I get my parts from ebay or perhaps someone from QB family might have spares. Good Luck and Enjoy!
Jon