Creating a portable library to run on both linux and windows

This is a typical way to export a DLL API for Windows and still support Linux:

#ifdef __cplusplus
extern "C" {
#endif

#ifdef _WIN32
#  ifdef MODULE_API_EXPORTS
#    define MODULE_API __declspec(dllexport)
#  else
#    define MODULE_API __declspec(dllimport)
#  endif
#else
#  define MODULE_API
#endif

MODULE_API int module_init();

#ifdef __cplusplus
}
#endif

In the DLL source:

#define MODULE_API_EXPORTS
#include "module.h"

MODULE_API int module_init()
{
    /* do something useful */
    return 0;
}

Your application source is correct.

Using the above model, on Windows the DLL will export the API while the application will import it. If not on Win32, the __declspec decoration is removed.

Since the header wraps the entire interface in extern "C", using the EXTERN_C macro on each interface is not required. extern "C" is used to tell the linker to use C linkage instead of C++. C linkage is standard across compilers, whereas C++ is not, limiting the use of a DLL to application built with the same compiler.

There is no need to integrate the return type into the API macro.