As a first step you might want to try gdbserver: Remote debugging with Eclipse CDT or Better variable exploring when debugging C++ code with Eclipse/CDT or Eclipse cdt remote build and remote debug on different servers offer advice on that.
Depending on your linux distribution, a hudson server can be set up as Computer B.
Hudson supports
- Ubuntu, Debian, Oracle Enterprise Linux, RedHat, CentOS, Fedora, openSUSE
as can be seen here.
Hudson has been developed in Java, but it is language agnostic, so it works for C as well.
In general, you can commit your changes (from Comp A directly or to a central Git Server) and Hudson (running on Comp B) will build it and deliver it to your debug server (Comp C), on which you are running GDB.
I lack experience with CDT, so I can't give you a more detailed instruction on how to set it all up apart from linking to manuals. And you'd have to test if GDB supports the Windows/Linux mix.