windows here, server there

I want to run a server program on a Redhat Linux 3.0ES host (which I've 
rented from RimuHosting.com) and have a Java Swing management console 
for that server which I can run at home on my Windows XP.  I can do most 
of this: with Cygwin.com on my home machine I can use X-windows 
forwarding and display Swing windows here with the program running there.

But I turn off my home machine every night, and I want the server 
program to run continuously.  This is the part I need help with.  How 
can I run a Java program on the remote Linux machine which will run 
continuously and also will be the client for my X-windows server at 
home, whenever I connect, for as long as I connect?

Thank you,
Rich Hammer