Depends on the router. Cisco is famously runs IOS, which I believe is based on BSD (FreeBSD 2.2 specifically). Barracuda gear, however, is Linux based.
However in all cases, I think you'd still be right about Bash not being present.
IOS-XE and XR are linux boxes, as are the nexus. Arista is linux based, Juniper is FreeBSD and Force10 is NetBSD (at least the older one were). Many devices in the class of enterprise or service provider grade are not going to be obtaining their addresses via DHCP and have a heavily modified version of the OS, anyway. In addition, best practices should be used in armoring the devices in this class, meaning filtering of the control plane, etc.
IOS XR used to be QNX though, if I'm not mistaken?
I hadn't heard of these other Cisco products until yesterday, but from what I read IOS XE and NX OS also run on Linux.
Anyhow, getting back to my original point about FreeBSD, there's a few sources that have linked IOS to FreeBSD. But after doing some digging of my own on this topic, I've come to the conclusion that this is one of those urban legends that's proliferated for whatever reason. So it would seem that you're right about IOS being home rolled. Interestingly though, they do have other products which are FreeBSD powered (eg AsyncOS http://www.cisco.com/c/en/us/products/security/email-securit...) but, as you said, not IOS
However in all cases, I think you'd still be right about Bash not being present.
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