Without Java or Flash, "running in a browser" is not exactly possible. There are solutions like (haven't tried this, was a top Google result):
but that isn't an SSH client in a browser, it's a browser-based interface to the SSH client on the server. Using it with key-based authentication means the private keys will need to be available to the *server*, which may be a deal-breaker if you're trying to build a service for people to connect to third-party SSH hosts.
There are similar projects for RDP:
But again, this is a "JS frontend to server-installed software" approach, not entirely client-side.
--Matt Jones