TL;DR: OS X is a Unix, while iOS is merely Unix-like but cannot, due to legal reasons, be called a Unix. Neither are based on Linux, which is itself another Unix-like but not a Unix.

Unix is a complicated concept and it has embodied different things in the past decades. It started out simply enough as a single body of code (with various additional patches made by different groups using it) and a trademark on the name at AT&T in the '70s. AT&T partnered with various educational institutions to help explore the concepts in Unix. The most important of these educational institutions was University of California, Berkeley which created its own patches for Unix and called them BSD ("Berkeley Software Distribution"). In the early '80s, AT&T decided it could make money by licensing the code out to commercial third parties, giving sale and binary redistribution rights. The many licensees (major ones were IBM, HP, Sun, and Microsoft) added additional code to their versions of Unix, much of which implemented similar features but which were all slightly incompatible with each other. In order to reign in the incompatibilities, various standards were created, such as the various versions of POSIX, and a Unix vendor could pay to be certified as meeting one of these standards. In the early '90s, AT&T sued Berkeley so Berkeley decided to rewrite all the copyrighted parts of BSD with freely-licensed reimplementations. Later in the '90s, AT&T sold the rights to the Unix intellectual property to Novell, who eventually transferred it to X/Open Consortium, who later merged with OSF and ultimately became the Open Group, where it remains today[1].

GNU/Linux is an operating system created in the early '90s that was heavily inspired by Unix (we say it is "Unix-like"), but which does not use any copyrighted Unix code and which is not certified as meeting any of the Unix compatibility standards. Due to its free nature, a belief that it was "good enough" for various commercial purposes, and its avoidance of violating the Unix copyright, it became a nice alternative to the licensed Unixes in the '90s, since many of those cost money, or were then in legally ambiguous situations.[2] The Linux kernel itself has continued to grow over the years and is now used for various operating systems beyond just GNU/Linux, the biggest of which is probably Android. As far as I know, none of the Linux-based operating systems today are certified by the Open Group.

Meanwhile, Mac OS X is based partially on BSD, which was not considered an official Unix, especially after AT&T's lawsuit. At some point, Apple decided to pursue Mac OS X compliance with the Open Group's UNIX 03 standard, which just required meeting a lot of technical requirements, and first got it with Mac OS X 10.5 Leopard. This makes Mac OS X an official Unix. El Capitan retains the UNIX 03 certification.[3] Since iOS, watchOS, and tvOS are based on Mac, they are Unix-like. But they're missing a lot of the requirements from any of the official Unix standards and so cannot be legally called Unixes.

Also, since none of Apple's operating systems use the Linux kernel it would be incorrect to say they were a flavor of Linux.

Hope this helps!

[1] https://en.wikipedia.org/wiki/History_of_Unix

[2] "Just for Fun" by Linus Torvalds. http://www.amazon.com/dp/0066620732

[3] http://www.opengroup.org/openbrand/register/brand3612.htm