Nonono.. they're not UNIX based at all. They're "unix-like".
There's a huge difference!
UNIX was an OS designed for those big computers in the 70s. It isn't in use (as far as I know) today.
It defined a number of "rules" of how operating systems should work, and these have been incorporated into the various unix-like (or *nix) systems.
Linux came about when Linus Torvalds decided that Minix, a unix-like OS, had a number of flaws - so decided to have a go at making his own OS, purely as a hobby.
Here's a timeline: http://www.levenez.com/unix/