how do we configure the TeamCity to build using Cygwin ?

Hello,

You could run a shell script with Cygwin by using the command line runner (https://www.jetbrains.com/help/teamcity/command-line.html) and setting up custom script which will call Cygwin bash.exe and pass in the shell script as a parameter. Please see this StackOverflow article for more details: https://stackoverflow.com/questions/22959618/how-to-run-cygwin-script-in-powershell (it is mentioning how to embed Cygwin call in Powershell, but the same method should work with the TeamCity).