I'm not sure about ZigBee PRO stack, but building MAC based application is fine also in Linux and macOS system.
By seeing makefile from ZigBee PRO, it seems difficult because some tools are provided only in Windows .exe file. The rough procedure is as below:
1. download toolchain (refer to JonasH's post.)
2. download JN-SW-4168 from NXP web site. This archive contains an exe file, but it can be unzipped.
3. install #1 and #2 at your Linux system.
note: the toolchain location is defined at makefiles from SDK. you may need to adjust it. In the $(SDK_BASE_DIR)/Chip/Common/Build/config.mk, toolchains PATH in the makefile is defined as below. Re-define or edit them to match your installation.
TOOL_BASE_DIR ?= $(SDK_BASE_DIR)/Tools
TOOL_COMMON_BASE_DIR ?= $(SDK_BASE_DIR)/../Tools
4. download application notes (e.g. AN1184) and extract them.
5. tailor makefiles in the example.
important part is SDK_BASE_DIR. This should be the path to the #2.
SDK_BASE_DIR = ../../../../sdk/$(JENNIC_SDK)
However $(OSCONFIG) $(PDUMCONFIG) $(ZPSCONFIG) in the makefile is the windows tools that will generate codes. This is the most challenging issue of the Linux environment. You may need to make an alternative command (or make these generated file from windows).
Also this makefile includes other makefiles from SDK directory and these makefiles might not consider about case sensitive filesystem. If the case, you need to edit these files if you encounter an build error.