Well, if you don't want them to have to install Cygwin, then that's a
bigger issue than just licensing. Think of Cygwin like an OS. If you
want to create something that can run under Windows, not Cygwin, then
you have to build it for Windows, not Cygwin. I don't know that it is
even possible to simply "bundle" Cygwin with your application. Cygwin
isn't just some little collection of libraries or something. It's a
whole system that must be correctly installed on someone's computer.
If you really want Mingw (a free compiler and development environment
for Windows), maybe what you should do is just download and install
Mingw, and use that, instead of doing it through the Cygwin compiler
using a barely-supported option. (Then you should get help with any
problems you have over at Mingw's website instead of here.)