To install Busybox in OSX I am thinking initramfs -filesystem in this thread.
I run unsuccessfully brew search busybox and no including package in Google clearly. 
Busybox is included in Linux by default but not in OSX. 
Different internals so not necessarily possible in OSX. 

How can you install busybox in OSX?
initramfs is something that is very specific to the Linux boot process (it's designed to make the kernel module loading process as flexible as possible) - you won't find an equivalent for OS X.

BusyBox, however, is available for Mac OS - see this Github repository for example.

That said, I'm not sure what problem you're trying to solve as Busybox is simply a set of small utilities that provide core Unix command line functions (ls, cp, etc). It's commonly used in boot environments and embedded systems due its wide platform support and small size.