You are almost there! Here is my solution, I am sure there are other ways to achieve this. Note that the solution assumes that your cygwin installation included the mintty terminal emulator (installed by default in recent versions of cygwin).
1) Right-click on the .minttyrc button and save the link as a text document on your computer.
2) Open a text document and copy its contents to the clipboard. A file might look something like this:
BackgroundColour=13,25,38 ForegroundColour=255,255,255 CursorColour=255,255,255 Black=0,0,0 BoldBlack=62,62,62 Red=217,20,46 BoldRed=246,162,173 Green=46,217,20 BoldGreen=173,246,162 Yellow=217,191,20 BoldYellow=246,235,162 Blue=20,46,217 BoldBlue=162,173,246 Magenta=191,20,217 BoldMagenta=235,162,246 Cyan=20,217,191 BoldCyan=162,246,235 White=217,217,217 BoldWhite=255,255,255
3) Open cygwin and go to your home directory by typing cd ~
4) Type ls -a to see various hidden configuration files that are stored in your home directory. In particular, you should see a file named .minttyrc .
5) Open .minttyrc with your favorite text editor and copy the code snippet that you received in step 2)
6) Restart cygwin or start another instance of the terminal, and the new color scheme should work.