Can I permanently delete the hard drive on a mac mini?

Clickhereand follow the instructions. If the computer originally shipped with Mac OS X 10.6.8 or earlier, when you reach step 5, insert its original disk, restart with the C or Option key held down, use the Disk Utility to erase the internal drive, and install a fresh OS.

If you're erasing a hard drive(not a SSD), choose to do so securely.

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Yes, this can be done.

Start by going into recovery mode >>> OS X: About OS X Recovery - Apple Support

Use disk utility to securely erase the mini's hd.

http://www.macworld.com/article/2906499/mac-911-how-to-erase-your-macs-hard-driv e-the-right-way.html

You are probably safe this time, but be cautious about advising "secure erase" regarding until you know it is NOT an SSDs. SSDs are very different from HD. HDs can "overwrite many times" without risk. When you overwrite completely an SSD you are coming close to if not actually choking the write-buffer of the SSD, and can kill the SSD (yes -- this is an occurrence of "software breaking hardware").

thanks for that... but that point is well covered in the Macworld article I referenced.