ci: Install cygwin on our Docker container

We install cygwin exactly as Kokoro does on their own containers.

.kokoro/gcp_windows_docker/Dockerfile

@@ -5,13 +5,16 @@
 # We build in .NET 6 so we need to install .NET 6 after.
 FROM mcr.microsoft.com/dotnet/framework/runtime:4.8-windowsservercore-ltsc2019
 
-# Setup Powershell
-RUN powershell -Command $ErrorActionPreference = 'Stop';
-RUN powershell -Command $ProgressPreference = 'SilentlyContinue';
+# Set default powershell policy for this script (ProgressPreference='SilentlyContinue' makes
+# downloads with Invoke-WebRequest not show the progress bar and is MUCH faster).
+SHELL ["powershell.exe", "-ExecutionPolicy", "Bypass", "-Command", "$ErrorActionPreference='Stop'; $ProgressPreference='SilentlyContinue'; $VerbosePreference = 'Continue';"]
 
 # Install Chocolatey
 ENV ChocolateyUseWindowsCompression false
-RUN powershell -NoProfile -ExecutionPolicy Bypass -Command "iex ((New-Object System.Net.WebClient).DownloadString('https://chocolatey.org/install.ps1'))" && SET "PATH=%PATH%;%ALLUSERSPROFILE%\chocolatey\bin"
+RUN Invoke-Expression ((New-Object System.Net.WebClient).DownloadString('https://chocolatey.org/install.ps1')); \
+    $old_path = [Environment]::GetEnvironmentVariable(\"PATH\", \"Machine\"); \
+    $all_users_profile = [Environment]::GetEnvironmentVariable(\"ALLUSERSPROFILE\", \"Machine\"); \
+    [Environment]::SetEnvironmentVariable(\"PATH\", $old_path + \";\" + $all_users_profile + \"\chocolatey\bin\", \"Machine\");
 
 # Install .NET Core 3.0 which we need for Google.Apis.Auth.AspNetCore3
 RUN choco install -y --no-progress dotnetcore-3.0-sdk
@@ -22,6 +25,16 @@ RUN choco install -y --no-progress dotnet-6.0-sdk
 # Install Git
 RUN choco install -y --no-progress git
 
+# Install cygwin and packages exactly as Kokoro does on its own containers
+RUN Invoke-WebRequest 'https://cygwin.com/setup-x86_64.exe' -OutFile C:/cygwin_setup.exe -UseBasicParsing; \
+    Start-Process 'C:/cygwin_setup.exe' -ArgumentList '-X', '--quiet-mode', '--root', 'C:\Cygwin64', '--site', 'http://ctm.crouchingtigerhiddenfruitbat.org/pub/cygwin/circa/64bit/2022/11/23/063529' -Wait -NoNewWindow; \
+    Remove-Item C:/cygwin_setup.exe; \
+    $old_path = [Environment]::GetEnvironmentVariable(\"PATH\", \"Machine\"); \
+    [Environment]::SetEnvironmentVariable(\"PATH\", $old_path + \";C:\Cygwin64\bin\", \"Machine\");
+
+# Restore default shell for Windows containers.
+SHELL ["cmd.exe", "/s", "/c"]
+
 # Default to PowerShell.
 # This is what Kokoro owned Docker images do. We'll do the same at least for now.
 CMD ["powershell.exe", "-NoLogo", "-ExecutionPolicy", "Bypass"]