There is really no differentiation within Firefox between what you (and I) would call an "upgrade" and an update, which one you are offered depends upon the setting Mozilla has set on the updates server. When automatic Firefox updates are turned on for Firefox, it will contact the update server once a day to see if an update is available. You can turn off that daily check, and thus turn off automatic update notifications.
https://support.mozilla.org/en-US/kb/Updating%20Firefox#w_how-do-i-configure-update-options
Automatically check for updates to: = uncheck Firefox

You can use the Help > Check for Updates menu item to manually check for an update to a newer 3.6.x version if one was released, but you will be offered an upgrade to the latest version with the 2nd and subsequent "check" when no update was found.

That said, Firefox 3.6.x will be End of Life on April 24. From then on there will be no more security patches. After that you will need to get beyond whatever problems you have with the later versions of Firefox or look for a different browser to use.

@the-edmeister - Thanks, so what I read from that answer is that there is a shortcoming in the design of the application and it fails to be able to provide a constrained test for upgrade/update by at least comparing two strings less than 5 characters in length (so far really the strings are only 2 characters - e.g.: 3 vs 10 but with the current version number style then 10 will be 99 in a short time.)

With the EOL of 3.6 then this problem will probably go away I guess - then I will have the last version of the quality Mozilla I have loved and will have to use it or Chromium needs to be installed since privacy was designed out of Mozilla and Chrome.

Gotta love having the ability to watch every access to every file by every application and see what actually takes place as opposed to that which is advertised.