According to wikipedia article on status codes, Nginx has a custom error code when http traffic is sent to https port(error code 497)

And according to nginx docs on error_page, you can define a URI that will be shown for a specific error.
Thus we can create a uri that clients will be sent to when error code 497 is raised.

nginx.conf

#lets assume your IP address is 89.89.89.89 and also that you want nginx to listen on port 7000 and your app is running on port 3000

server {
    listen 7000 ssl;
 
    ssl_certificate /path/to/ssl_certificate.cer;
    ssl_certificate_key /path/to/ssl_certificate_key.key;
    ssl_client_certificate /path/to/ssl_client_certificate.cer;

    error_page 497 301 =307 https://89.89.89.89:7000$request_uri;

    location / {
        proxy_pass http://89.89.89.89:3000/;

        proxy_pass_header Server;
        proxy_set_header Host $http_host;
        proxy_redirect off;
        proxy_set_header X-Real-IP $remote_addr;
        proxy_set_header X-Forwarded-Protocol $scheme;
    }
}

However if a client makes a request via any other method except a GET, that request will be turned into a GET. Thus to preserve the request method that the client came in via; we use error processing redirects as shown in nginx docs on error_page

And thats why we use the 301 =307 redirect.

Using the nginx.conf file shown here, we are able to have http and https listen in on the same port

There are two parts to getting this to work, so I'll address them seperately.

Connecting to your Server:

As you've mentioned that all you can talk to is a proxy on 443 and 80, you'll need to tunnel your SSH connection out through the proxy. You do that by telling Putty to use the proxy server to connect. Under the 'proxy' menu. Select HTTP and then enter the details for the corporate proxy.

From your post, it seems like you've got the connection working fine.

Configure Putty to Create a SOCKS proxy on the local machine

Both putty and OpenSSH support opening a SOCKS proxy. For OpenSSH you'd use:

ssh -D <port>

And then point your browser at that port. In putty you create a 'dynamic' port forward. You'll find it under the Tunnels menu. Enter your desired listening port and then enter anything you like as a destination (it gets ignored for dynamic forwards). You can then point your browsers proxy settings at that localhost:<port> and it should work.

For more information, the relevant part of the putty documentation is at http://tartarus.org/~simon/putty-snapshots/htmldoc/Chapter3.html#using-port-forwarding.

As others have posted you may not want to circumvent the corporate filtering though as it's not something you may want to explain :).