I had the same problem, that is, after upgrading from Mavericks to Yosemite I got the following error: nginx: [emerg] mkdir() "/usr/local/var/run/nginx/client_body_temp" failed (2: No such file or directory)
All I needed to do to solve this issue was to create the folder:
mkdir -p /usr/local/var/run/nginx/client_body_temp