Okay, I'll throw out a few ideas and I'll let you pick which one you may think is easiest.
1. Use the internet toolkit ($$$) or build a simple telnet client yourself in LabVIEW (use the TCP VIs in labview). This would allow you to connect to the telnet port of your linux box, login, password, and execute a command. (You'll probably want to use nohup, do a man on nohup to find out more).
2. Install SSH daemon on your linux box, and an ssh client on your windows machine. You can't get SSH for windows from cygwin. This would be more secure, and you wouldn't need to use the TCP VIs, you could simply use the "system exec" vi to call SSH.
3. On your linux box, install your app in inetd. Then, whenever someone connected to a certai
n port # on the linux box, it would execute your command. A little dangerous because someone might try to connect 50 million times (or a port scan) and kill your box. You also wouldn't be able to send a dynamic command line argument.
4. Write a small TCP application on your linux box to wait for connections (listen for command line options) and execute your command.
You'll probably find one of these options ideal depending on your administration and programming background.