Calibre added support to Auth via header in v0.6.5. You need to set the Calibre settings in the Admin Configuration.
location /calibre {
proxy_bind $server_addr;
proxy_pass http://127.0.0.1:8083;
proxy_set_header Host $http_host;
proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
proxy_set_header X-Scheme $scheme;
proxy_set_header X-Script-Name /calibre; # IMPORTANT: path has NO trailing slash
auth_request /organizr-auth/4; #Change the X to whatever group you want to allow access
auth_request_set $auth_user $upstream_http_x_organizr_user;
proxy_set_header X-WEBAUTH-USER $auth_user;
}