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In addition to Dan Williams' answer, if you want to add a Startup/Shutdown script, you need to be looking for Windows Settings under Computer Configuration. If you want to add a Logon/Logoff script, you need to be looking for Windows Settings under User Configuration.

So to reiterate what Dan said with this information included,

For Startup/Shutdown:

1. Run gpedit.msc (Local Policies)


2. 
   Computer Configuration -> Windows Settings -> Scripts -> Startup or Shutdown -> Properties -> Add

For Logon/Logoff:

1. Run gpedit.msc (Local Policies)


2. 
   User Configuration -> Windows Settings -> Scripts -> Logon or Logoff -> Properties -> Add

Source: http://technet.microsoft.com/en-us/library/cc739591(WS.10).aspx

For those who prefer using the Task Scheduler, it's possible to schedule a task to run after a restart / shutdown has been initiated by setting the task to run after event 1074 in the System log in the Event Viewer has been logged.
However, it's only good for very short task, which will run as long as the system is restarting / shutting down, which is usually only a few seconds.

• From the Task Scheduler:
  

  
  

  Begin the task: On an event
  
  Log: System
  
  Source: USER32
  
  EventID: 1074

  
  

  
  


• From the command prompt:
  

  
  

  
  

  schtasks /create /tn "taskname" /tr "task file" /sc onevent /ec system /mo *[system/eventid=1074]

  
  

  
  

  
  

Comment: the /ec option is available from Windows Vista and above. (thank you @t2d)

Please note that the task status can be:

The operation being requested was not performed because the user has not logged on to the network. The specified service does not exist. (0x800704DD)

However, it doesn't mean that it didn't run.

One workaround might be to write a simple batch file to run the program then shutdown the computer.

You can shut down from the command line -- so your script could be fairly simple:

c:directorymyProgram.exe
C:WINDOWSsystem32shutdown.exe -s -f -t 0

If you run GPEdit.MSC you can go to Computer Configuration -> Windows Settings -> Scripts, and add startup /shutdown scripts. These can be simple batch files, or even full blown EXEs. Also you can adjust user configurations for logon and logoff scripts in this same tool. This tool is not available in WIndows XP Home.

The Group Policy editor is not mentioned in the post above. I have used GPedit quite a few times to perform a task on bootup or shutdown. Here are Microsoft's instructions on how to access and maneuver GPedit.

How To Use the Group Policy Editor to Manage Local Computer Policy in Windows XP

You can run a batch file that calls your program, check out the discussion here for how to do it:
http://www.pcworld.com/article/115628/windows_tips_make_windows_start_and_stop_the_way_you_want.html

(from google search: windows schedule task run at shut down)

What I can suggest doing is creating a shortcut to the .bat file (for example on your desktop) and a when you want to shutdown your computer (and run the .bat file) click on the shortcut you created. After doing this, edit the .bat file and add this line of code to the end or where needed:

c:windowssystem32shutdown -s -f -t 00

What this does it is

1. Runs the shutdown process


2. Displays a alert


3. Forces all running processes to stop


4. Executes immediately

On Windows 10 Pro, the batch file can be registered; the workaround of registering cmd.exe and specifying the bat file as a param isn't needed. I just did this, registering both a shutdown script and a startup (boot) script, and it worked.