you can find most of the public information on a specific server by going to its website at http://a.root-servers.org/ (replace a with any letter from a to m) as well as their FAQ.
The most relevant part is "Q: Yes but what if there is a software bug?
A: There is significant software diversity across the root name servers. There are at least three major DNS implementations being used: bind8, bind9 and NSD. The servers also run on very diverse hardware and operating system platforms. Actually maintaining this diversity is a major part of the coordination going on between root name server operators. "
So no, chances are they don't all run linux, and certainly not the same distro