Can I stop the "this website does not supply identity information" message
"This website does not supply identity information." is not only about the encryption of the link to the website itself but also the identification of the operators/owners of the website - just like it actually says. For that warning (it's not really an error) to stop, I believe you have to apply for the Extended Validation Certificate https://en.wikipedia.org/wiki/Extended_Validation_Certificate. EVC rigorously validates the entity behind the website not just the website itself.
Firefox shows the message
"This website does not supply identity information."
while hovering or clicking the favicon (Site Identity Button) when
- you requested a page over HTTP
- you requested a page over HTTPS, but the page contains mixed passive content
HTTP
HTTP connections generally don't supply any reliable identity information to the browser. That's normal. HTTP was designed to simply transmit data, not to secure the data it transmits.
On server side you could only avoid that message, if the server would start using a SSL certificate and the code of the page would be changed to exclusively use HTTPS requests.
To avoid the message on client side you could enter about:config in the address bar, confirm you'll be careful and set browser.chrome.toolbar_tips = false.
HTTPS, mixed passive content
When you request a page over HTTPS from a site which is using a SSL certificate, the site does supply identity information to the browser and normally the message wouldn't appear.
But if the requested page embeds at least one <img>, <video>, <audio> or <object> element which includes content over HTTP (which won't supply identity information), than you'll get a so-called mixed passive content * situation.
Firefox won't block mixed passive content by default, but only show said message to warn the user.
To avoid this on server side, you'd first need to identify which requests are producing mixed content.
With Firefox on Windows you can use Ctrl+Shift+K (Control-Option-K on Mac) to open the web console, deactivate the css, js and security filters, and press F5 to reload the page, to show all the requests of the page.
Then fix your code for each line which is showing "mixed content", i.e. change the appropriate parts of your code to use https:// or, depending on your case, protocol-relative URLs.
If the external site an element is requested from doesn't use a SSL certificate, the only chance to avoid the message would be to copy the external content over to your site so your code can refer to it locally via HTTPS.
* Firefox also knows mixed active content, which is blocked by default, but that's another story.