Nginx does not have a way to turn off the server header, the closest option is the server token directive but this only turns off the version number.
I have never actually tried this, but the proxy_pass_header directive might allow you to specify the Server header as something that should be passed on to the client.
I'd personally recommend sending the machine ID in a custom header designed for it (X-WHATEVER) as using a RFC defined header for behaviour other than what it's designed for is bound to bite you eventually.
If proxy_pass_header doesn't work then I'm pretty sure the only alternative is that you patch the Nginx source to not overwrite it.