Does cygwin allow a statically compiled binary? This would prevent the need for cygwin1.dll being on the PATH of target machines.
Cygwin's GCC no longer supports -mno-cygwin.
-mno-cygwin
Try to use mingw-gcc instead, e.g. x86_64-w64-mingw32-gcc, as mentioned in the Cygwin FAQ.
x86_64-w64-mingw32-gcc