Cygwin on a Windows 7 Box.
Start Cygwin Bash Shell and issue startxdmcp command:

startxdmcp.bat 10.12.13.14
# where 10.12.13.14 is the IP of the remote Linux Box
# you are trying to get a login window from

# try it raw aswell, no change
XWin.exe -ac -query 10.12.13.14

Problem:
Windows 7 Cygwin XWin displays black / blank screen

Resolution:
Make sure the DNS Record for you local computer is correct

rem Find out what the dns server thinks the IP Address is
C:\Users\jmuser>nslookup mypc
Server:  dnsserver.home.local
Address:  10.12.13.4

Name:    mypc.home.local
Address:  10.12.13.20
REM This differs from what the actual address is 10.12.13.107

REM Issue ipconfig command to re-register the workstations
REM DNS Record
REM You have to be an administrator to do this I think.
C:\Users\jmuser>ipconfig /registerdns

Windows IP Configuration

Registration of the DNS resource records for all adapters of this computer has b
een initiated. Any errors will be reported in the Event Viewer in 15 minutes.

REM Check that it's worked.
C:\Users\jmuser>nslookup mypc
Server:  dnsserver.home.local
Address:  10.12.13.4

Name:    mypc.home.local
Address:  10.12.13.107

So as far as I can tell if the DNS record for you XDMCP client is wrong the remote Linux computer serving the XDMCP login screen will resolve the wrong IP and the login screen won't appear on the local computer.

startxdmcp.bat 10.12.13.11

# working now