Looks like your provider doesnt like the way you're pulling your setup files?

[ghost]

Possible corrupt mirror?  Setup.ini rejected.
           unattended_mode is set at mbox: returning default value
           mbox note: Unable to get setup from <http://mirrors.kernel.org/sourceware/cygwin/>
           unattended_mode is set at mbox: returning default value
           download/verify error in unattended_mode: out of retries
           note: Installation incomplete.  Check C:\cygwin\var\log\setup.log.full for details
           Ending cygwin install
           STDERR:
           ---- End output of setup.exe -q -O -R C:/cygwin -s http://mirrors.kernel.org/sourceware/cygwin/  ----
           Ran setup.exe -q -O -R C:/cygwin -s http://mirrors.kernel.org/sourceware/cygwin/  returned 1

           Resource Declaration:
           ---------------------
           # In C:\Users\vagrant\AppData\Local\Temp\kitchen\cookbooks\cygwin\recipes\default.rb

            28: execute "setup.exe" do
            29:   cwd node['cygwin']['download_path']
            30:   command "setup.exe -q -O -R #{node['cygwin']['home']} -s #{node['cygwin']['site']} #{proxy_command}"
            31:   not_if { File.exists?("C:/cygwin/etc/passwd") }
            32: end
            33:

           Compiled Resource:
           ------------------
           # Declared in C:\Users\vagrant\AppData\Local\Temp\kitchen\cookbooks\cygwin\recipes\default.rb:28:in `from_file'

           execute("setup.exe") do
             action [:run]
             retries 0
             retry_delay 2
             default_guard_interpreter :execute
             command "setup.exe -q -O -R C:/cygwin -s http://mirrors.kernel.org/sourceware/cygwin/ "
             backup 5
             cwd "C:\\Users\\vagrant\\AppData\\Local\\Temp\\kitchen\\cache"
             returns 0
             declared_type :execute
             cookbook_name :cygwin
             recipe_name "default"
             not_if { #code block }
           end

I was able to get it to converge by modifying your default recipe and changing the install command

execute "setup.exe" do
  cwd node['cygwin']['download_path']
  command "setup.exe -q -O -R #{node['cygwin']['home']} --no-desktop --download --local-install -s #{node['cygwin']['site']} #{proxy_command}"
  not_if { File.exists?("C:/cygwin/etc/passwd") }
end