I was wondering if there is an SSH client that will let me set up automated backups of windows computers to a Linux server via SSH. I am willing to pay for software if no free software will do it.

I use an rsync command in a .bat file on Cygwin. It's reliable and not too hard to set up.

Start at cygwin.com, and be sure to add the rsync and OpenSSH packages when you install. Note that you do not need an rsync server (rsyncd) running on the Linux machine; just an ssh server will work. You do need to have rsync installed on the Linux machine.

Your command should be something like:

rsync -av --progress -e ssh /cygdrive/c/ ssbn@ssbn.com:/home/ssbn/windows_backup

You could also try this prepackaged version of rsync:

www.itefix.no/cwrsync/

Though I haven't used it myself, Unison seems to be pretty popular:
www.cis.upenn.edu/~bcpierce/unison/