What is a good compatible external cd/dvd player burner for my mini Mac?

You mean, that's cheaper than Apple's offering? 😉

Any Samsung or other burner available from Amazon will work - just make sure that it can connect to your Mac Mini.

Check this company out. I purchasedt many hard drives from them in their a reliable company while well worth the price you pay for their hard drives.

http://www.newertech.com/products/ministackmax.php

I have been very happy with the Samsung Ultra Portable drive. And you can't beat the price!

Although not cheaper than Apple's Superdrive this driver from otherworldcomputing.com is blazingly fast in reading and ripping disks, quiet and is a drawer loader: OWC Mercury Pro 24X External DVD Burner. The reason I got this one is that it has Firewire since my iMac doesn't have USB 3.