There's only one book reference to Valyrian Steel being able to kill Others. and it's speculation. Dragon Steel, what ever it is, can, but it may or may not be Valyrian Steel. There's nothing firm. I just looked.
Some think dragon steel to be Valyrian steel, mostly because Jon and Sam speculate on it being so but some do not and think them separate things because dragon steel was mentioned as being used long before the Targs ever set foot on Westeros. Dragon glass and dragon steel, are mentioned separately, so I doubt they are the same.
It may be possible that Jon has a special ability or something. Who knows..