So, as someone who helped design the operating environment for the latest REAL Destroyers, the newest of the DDG-51 class, and who has an extreme level of experience with the systems running on the DDG-51s, CG-47s, and the LM version of the Littoral Combat System, the core of the system is, in fact, running a variant of Red Hat Enterprise Linux. The CDS consoles, at least on AEGIS (DDG-51, CG-47) run RHEL, as well, but there are input/output boards that we call 'ACEG' that, in certain instances, run LynxOS or VxWorks, and take data feeds from legacy sensors, then convert them for usage by the actual core operating environment.
There are also dozens of government furnished systems that interface with the combat systems, and many of those run as embedded systems with little to no true networking capability.