The thing is that, when you "rewrite" into URI having protocol and hostname (that is http://app.newexample.com/
in your case), Nginx issues fair HTTP redirect (I guess the code will be 301 aka "permanent redirect"). This leaves you only two mechanisms to transfer any information to the handler of new URL:
Since you are redirecting users to the new domain, cookie is no-go. But even in the case of a common domain I would choose URL to transfer this kind of information, like
server_name app.example.com;
location /app/ {
rewrite ^/app/(.*)$ http://app.newexample.com/$1?from_old=yes;
}
This gives you the freedom to process at either Nginx or in a browser (using JavaScript). You may even do what you wanted intially, issuing a special HTTP header for JavaScript in new app server Nginx configuration:
server_name app.newexample.com;
location /app {
if ($arg_from_old) {
add_header X-From-Old-Site yes;
}
}