are you sure the host is actually alive ?  what does your connection settings look like ?  is there anything useful in the JS console (ctrl+shift+j) ?

Yes, the host to which I am trying to connect is alive. I can connect to it from Debian and Ubuntu boxes on the same subnet as the Chromebook. 

I have not touched the settings. I have uninstalled the ssh extention, rebooted Chromebook, re-installed. Then I "ssh uname@fqdn" just as I do elsewhere.

Here is the JS console output.

init: hterm
nassh_deps.concat.js:190 init: nassh
nassh_main.js:39 Did you know there is a dedicated options page?
No need to mess with term_ directly anymore!
chrome-extension://pnhechapfaindjhompbnflcldabbghjo/html/nassh_preferences_editor.html
nassh_command_instance.js:636 Uncaught TypeError: Cannot read property 'match' of undefined
    at Function.nassh.CommandInstance.splitCommandLine (nassh_command_instance.js:636)
    at nassh.CommandInstance.connectTo (nassh_command_instance.js:791)
    at nassh.CommandInstance.connectToDestination (nassh_command_instance.js:579)
    at nassh.CommandInstance.connectToArgString (nassh_command_instance.js:392)
    at onFileSystemFound (nassh_command_instance.js:207)
    at getOrCreateNextName (nassh_deps.concat.js:18291)
nassh_command_instance.js:911 Uncaught TypeError: Cannot read property 'postMessage' of undefined
    at nassh.CommandInstance.sendToPlugin_ (nassh_command_instance.js:911)
    at nassh.CommandInstance.onTerminalResize_ (nassh_command_instance.js:930)
    at hterm.Terminal.IO.onTerminalResize_ (nassh_deps.concat.js:14253)
    at hterm.Terminal.realizeSize_ (nassh_deps.concat.js:11814)
    at hterm.Terminal.onResize_ (nassh_deps.concat.js:14048)
    at notifyList (nassh_deps.concat.js:8341)
nassh_command_instance.js:911 Uncaught TypeError: Cannot read property 'postMessage' of undefined
    at nassh.CommandInstance.sendToPlugin_ (nassh_command_instance.js:911)
    at nassh.CommandInstance.onTerminalResize_ (nassh_command_instance.js:930)
    at hterm.Terminal.IO.onTerminalResize_ (nassh_deps.concat.js:14253)
    at hterm.Terminal.realizeSize_ (nassh_deps.concat.js:11814)
    at hterm.Terminal.onResize_ (nassh_deps.concat.js:14048)
    at notifyList (nassh_deps.concat.js:8341)
nassh_command_instance.js:911 Uncaught TypeError: Cannot read property 'postMessage' of undefined
    at nassh.CommandInstance.sendToPlugin_ (nassh_command_instance.js:911)
    at nassh.CommandInstance.onTerminalResize_ (nassh_command_instance.js:930)
    at hterm.Terminal.IO.onTerminalResize_ (nassh_deps.concat.js:14253)
    at hterm.Terminal.realizeSize_ (nassh_deps.concat.js:11814)
    at hterm.Terminal.onResize_ (nassh_deps.concat.js:14048)
    at notifyList (nassh_deps.concat.js:8341)
nassh_command_instance.js:911 Uncaught TypeError: Cannot read property 'postMessage' of undefined
    at nassh.CommandInstance.sendToPlugin_ (nassh_command_instance.js:911)
    at nassh.CommandInstance.onTerminalResize_ (nassh_command_instance.js:930)
    at hterm.Terminal.IO.onTerminalResize_ (nassh_deps.concat.js:14253)
    at hterm.Terminal.realizeSize_ (nassh_deps.concat.js:11814)
    at hterm.Terminal.onResize_ (nassh_deps.concat.js:14048)
    at notifyList (nassh_deps.concat.js:8341)
nassh_command_instance.js:911 Uncaught TypeError: Cannot read property 'postMessage' of undefined
    at nassh.CommandInstance.sendToPlugin_ (nassh_command_instance.js:911)
    at nassh.CommandInstance.onTerminalResize_ (nassh_command_instance.js:930)
    at hterm.Terminal.IO.onTerminalResize_ (nassh_deps.concat.js:14253)
    at hterm.Terminal.realizeSize_ (nassh_deps.concat.js:11814)
    at hterm.Terminal.onResize_ (nassh_deps.concat.js:14048)
    at notifyList (nassh_deps.concat.js:8341)

what do you mean you "ssh uname@fqdn" ?  you type that into the omnibox ?

If I did try and connect literally to "uname@fqdn" then I would eventually get an error message, would I not? And not what I report. I appreciate you're trying to help, but I have ALREADY reported that I can ssh to the destination box from other non-Chromeboook boxes on the same subnet. The same symptom occurs to any box I try to ssh to, whether from the omnibox or the pop up connection box.

I am a practiced (decades long) user of ssh, and over a year of that on Chromebooks too. Nevertheless, please DO point out the silly mistake I am making. Or accept the bug report! 

I have now installed the Android App JuiceSSH on to the Chromebook. It connects no problem.

To be clear: The standard ssh extension/app has worked for me absolutely fine for a long time. It has stopped working. Before I reported this I downgraded from the Dev channel to the regular channel. Same problem. I have since reporting the issue upgraded to the beta channel. Same problem.

OK, more testing. Now on the Beta channel, so this may explain any difference in behaviour, I do not know. The pop-up box works, or has the last 10 times I tried, unlike before. If one types "ssh " in the ominbox then as usual the prompt changes. You type in a (valid) "uname@fqdn" (without the quotes) as would ordinarily work. Then the following is displayed:
----------------
Welcome to Secure Shell version 0.8.36.11.
Answers to frequently asked questions: https://goo.gl/muppJj (ctrl+click on links to open)

[Pro Tip] Use 'Open as Window' to keep Control-W from closing your terminal!
                                                                            [Pro Tip] See https://goo.gl/muppJj for more information.

ChangeLog/release notes: https://goo.gl/YnmXOs
Major changes since 0.8.35.6:
 ¤ SFTP mount support (Chrome OS only).
 ¤ OpenSSH upgraded to 7.5p1 (ECDSA/ED25519 support, but older features deprecated).
 ¤ Support for ssh:// links.
 ¤ Omnibox integration for ssh keyword.
 ¤ Add a user-css-text field for injecting custom CSS directly.
 ¤ IDN support for international domains.
 ¤ Root path for SFTP mounts can be configured (Chrome OS only).
 ¤ OpenSSL upgraded to 1.0.2k.
 ¤ Custom "mouse-paste-button" settings need to be manually updated.
 ¤ Mouse wheel sends up/down arrow keys on the alt screen in DECCKM mode.
 ¤   Hold shift to disable temporarily.
 ¤   New scroll-wheel-may-send-arrow-keys preference to enable.
 ¤ Support for iTerm2 OSC-9 growl notifications.
 ¤ Support for URxvt OSC-777:notify module notifications.
 ¤ Ctrl+V/Ctrl+Shift+V now pastes directly (fixes at least macOS).
 ¤ Support DOCS (ESC+percent) sequences to put into UTF8 mode.
 ¤   Can be used to disable graphics maps permanently.
 ¤ The SSH command line can handle basic quoting rules (e.g. -o "Feature yes").
 ¤ Unicode combining character processing has been overhauled.

Random Pro Tip #4: Chrome OS can mount filesystems via SFTP and browse via the Files app.  Use 'SFTP Mount' instead of 'Connect'.

Connecting to xxxx@xxxx...
-----------
and at that point it hangs, cursor on the line below, under the "C", indefinitely.

I see the same behavior.  If I open ssh from the Apps icon and enter the connection details (user@fqdn) I can connect.  But if I open a new browser window, then hit ssh <TAB> and enter the user@fqdn  it hangs in the same manner as described above.  With secure shell 0.8.36.11 Chrome Version 60.0.3112.101 (Official Build) (32-bit) on Windows 7

The following revision refers to this bug:
  https://chromium.googlesource.com/apps/libapps/+/0d2d3a5b554efeff4cace90f8c553201b31c5666

commit 0d2d3a5b554efeff4cace90f8c553201b31c5666
Author: Mike Frysinger <vapier@chromium.org>
Date: Tue Sep 12 17:42:04 2017

nassh: fix argstr parsing when connecting via URIs

We run through this code with undefined argstr when connecting via URIs
or the omnibox.  Add a fallback for when the input is undefined or null.

BUG= chromium:762484 

Change-Id: If9ebdca1278e2122017a6dfead53e0c32a0153b1
Reviewed-on: https://chromium-review.googlesource.com/656053
Reviewed-by: Brandon Gilmore <varz@google.com>
Tested-by: Mike Frysinger <vapier@chromium.org>

[modify] https://crrev.com/0d2d3a5b554efeff4cace90f8c553201b31c5666/nassh/js/nassh_command_instance_tests.js
[modify] https://crrev.com/0d2d3a5b554efeff4cace90f8c553201b31c5666/nassh/js/nassh_command_instance.js