VMware officially supports the Apple Mac Mini, see the VMware Compatibility Guide. However, since the Apple Mac Mini is not an Enterprise-class system but instead a consumer system without Error-Correcting Code (ECC) memory, issues including software crashes can occur. This document outlines some of the common issues that may occur when running ESXi on such consumer grade devices and outlines some basic steps that you can troubleshoot before contacting VMware Support.

VMware vSphere ESXi 6.5
VMware vSphere ESXi 6.7
VMware vSphere ESXi 6.0
VMware vSphere ESXi 7.0.0

During the system crash, if a machine check exception (MCE) is thrown and a purple diagnostic screen displays, a hardware problem has caused it. There is no other way to generate an MCE. You should work with their hardware vendor to replace the faulty system. If an MCE is not thrown and a purple diagnostic screen displays, it may still be related to a hardware issue and before contacting VMware Support, ensure you can consistently reproduce the issue on another system using the exact workflow which caused the initial system crash. 

Here are some basic troubleshooting steps that you should run through prior to contacting VMware Support:

1.  Ensure that you have the latest BIOS running on the Apple Mac Mini. To update to the latest BIOS, you will need to boot the Apple Mac Mini using Apple macOS and apply all patches to ensure the system is up to date. Not having the latest BIOS can cause system instability if there is a bug in the Firmware.
2. Ensure that macOS can run properly on the Apple Mac Mini system without any issues. In addition, you can also run the Apple Hardware Test to help diagnosis the underlying hardware. 
3. If ESXi is installed on a USB device, ensure that the USB device is not having any issues by replacing it with a new device or attempt to boot ESXi from local storage. 

Note: vSphere running on Apple hardware platform is an officially supported configuration from VMware. But it is not a published configuration from Apple Inc. If any support call results in perceived hardware issue, you are expected to resolve the issue directly with Apple and/or your hardware vendor. 

For more information, see Decoding Machine Check Exception (MCE) output after a purple screen error