I am looking for a means to connect, preferably securely, to a linux
or unix system running X Windows, directly from my Treo 600.  I
already have internet access on the phone.  I would prefer not to use
vnc because it would require leaving a window open even when I'm not
connected.  This should not preclude the use of a command line.

I think "Intellisync" would do this, have a look here

http://www.intellisync.com/go/products/client-synchronization/intellisync-handheld-edition/intellisync-trial/index.cfm

Good luck

I did not find any button to submit an answer, but it is the answer :)
Step one: Download and install
http://palmvnc2.free.fr/stable/PalmVNC.prc, VNC client for PalmOS
Step two: Install VNC server on your Linux machine -- (x11vnc)
http://www.karlrunge.com/x11vnc/
Step three: Configure and run VNC server, do not forget to set password!
Step four: Run PalmVNC from your favorite PalmOS launcher, connect and enjoy...

But... you will have a problem. PalmOS has a limitation on connection
speed, so your connection will be terribly slow. I would suggest you
to use http://www.sealiesoftware.com/pssh/dist/pssh-2004_12_30.prc and
(Open)SSH server (aka sshd) on your Linux PC. You won't see any X11
graphics but text mode and connection will be secure and really fast.