To install Busybox in OSX
Solution 1:
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.