I'm not sure, but I think you are asking about an SSH client for Kindle?
I got a very simple terminal going on my Kindle 2, when I had it (keyboard input, screen display). You could ssh from there, if you installed a precompiled binary. But it wasn't very useful.
I think web-based SSH actually could work for you, if you have sufficient access to the server to enable them. They are secure like SSH, so long as you use HTTPS and connect directly to the SSH server with the web browser. See the "Important issues", along with the list of available options at the bottom of this page: