When installing the Eclipse IDE, one must be running a version of the Java Virtual Machine (JVM) that is compatible with the version of Eclipse that is being installed.
As a result, upon trying to install Eclipse Neon, you will likely see a box that looks like this (except it will say 1.8 rather than 1.7):
Though there are multiple ways to upgrade Java on the Mac, it seems getting it from Oracle is the only one that works from what I’ve seen.
To upgrade the JVM on macOS to 1.8, go to the Oracle Java SE Runtime Environment downloads page and get the file for Mac; I prefer getting the dmg package file:
Install the Java SE package like any other Mac application and Eclipse can then be installed.