Ok, I think you misunderstood what I meant. Let me try to clarify this a bit.
I didn’t mean that you wrote any line of Obj-C code yourself. However, a lot of Apple APIs are written in Objective C. So JUCE as a cross-platform framework of course contains some Objective C code to implement the MacOS / iOS specific stuff. However JUCE only presents you a C++ interface.
Now how does this work? Beneath pure C++ sources (.cpp) and pure Objective C sources (.m) clang (the compiler used for Mac and iOS) also allows to compile Objective C++ sources (.mm). Objective C++ sources can contain Objective C code as well as C++ code or both. So all modules that come with JUCE usually contain a .mm file in its root directory beneath the .cpp file that do nothing more than just include the .cpp file (take a look here for an example: https://github.com/WeAreROLI/JUCE/blob/develop/modules/juce_audio_devices/juce_audio_devices.mm). Now if the Projucer sees a .mm file and a .cpp file in the module root folder, it decides to compile the .mm file as Objective C++ source and just ignore the C++ file. This allows JUCE to include Apples Objective C headers somewhere deep down in the framework when they are needed to talk to the OS.
This was just to explain what I meant.
Now regarding your actual problem, I wanted to point you to the fact that if you include some headers in your .cpp file that somewhere deep down the include path includes an Objective C header this might not work if your file is called .cpp as clang then tries to compile it as pure C++ source. Trying to do so leads to error messages exactly like the one you posted. You very likely won’t encounter this problem with a Windows build as this includes some other headers under the hood - this is just how the encapsulation of the OS specific stuff works. This could be solved by adding a .mm file that just includes the .cpp file. It will be ignored on non-apple systems.
However, if you encounter this problem with one of the JUCE demo projects the source of the problem might obviously not be your own code.
I hope this helped you to get a better idea of what I meant and gives you some hints to search for the solution to your problem.