Hi,
try to specify “/cygdrive/c/root/include” as include path instead of “C:\root\include”. And I believe your ROOT build was with the Microsoft Visual C++ compiler, while you are trying to build with cygwin GCC. Those two are not compatible. You should probably use MS DevStudio Express which costs 0$ instead of Eclipse; check Google for where to get it.
Cheers, Axel.