If you look in your cache directory and the css an js files are not there, most likely you have a mod_rewrite problem.
Until you get it sorted out, There is an option in the Minify section that does not use mod_rewrite to create the files, it directly calls the minify.php file.
Rewrite URL structure
If disabled, CSS and JS embeddings will use GET variables instead of "fancy" links.
If after unchecking that box the files start showing up in your cache directory then you know that W3TC can create the files and write them.
When you check that option, mod_rewrite will point to the minify.php file.
RewriteRule ^(.+\.(css|js))$ ../../plugins/w3-total-cache/pub/minify.php?file=$1 [L]
So basically you need to make sure that your server is processing the .htaccess file and that it's there. If you don't process .htaccess files then it needs to be in your virtual host, same with ngnix.