These config settings work for me:
{
// start bash, not the mintty, or you'll get a new window
"terminal.integrated.shell.windows": "C:\\cygwin\\bin\\bash.exe",
// Use this to keep bash from doing a 'cd ${HOME}'
"terminal.integrated.env.windows": {
"CHERE_INVOKING": "1"
},
// Make it a login shell
"terminal.integrated.shellArgs.windows": [
"-l"
],
}