Swift doesn´t add specific rewrite rules automatically to enable rules on NGINX as it would do on an Apache server in .htaccess.
You can view the generated rules in Swift dashboard here
These rules you need to add to your NGINX.conf. However, most (NGINX) servers are already configured to work out of the box regarding GZIP, Browser cache, expire headers, etc.. So probably, you are already good to go.
Most common issue with NGINX and Swift is double GZIP. If after activating Swift you notice pages are completely broken (cryptic symbols) you need to disable Enable Gzip in Settings->Caching->General.
Note. After you made changes to your NGINX.conf, always test your Nginx configuration and then reload it.