Your guess is correct - you need newer headers.
A bit of explanation:
If you use Windows, you don't get new API 'out of the box'. Headers provided with Windows SDK and SDK included in Visual Studio has only standard GL.h, containing only functions present in MS generic implementation (oooold).
Basically, when you create apps using OpenGL, you need to download the newest header from OpenGL site and then query adresses of required functions directly from DLL that contains OpenGL implementation - wglGetProcAddress(). The reason for that, is because GL implementation is provided by GPU vendors and updated automatically when GPU drivers are being installed - that's also the reason, why GPU drivers updating is so important when it comes to OpenGL.
But it's a mess. And that's why various libraries were developed, that allows you to make it in more automated way. The best and most-widely used example is GLEW.
You already use it, so you probably know some of the above.
Now, the problem is probably with version of GLEW you're using now - it simply doesn't define some constants, so you are not able to use them. If I remeber correctly, support for compute shaders was added in version 1.7.0. It IS supported by now, beacause I use it in my project.
Simply go to download page of GLEW and get the newest version. Replace current headers, so new features will be visible.
Also, don't forget to replace .lib files and copy DLL's to Bin folder of your project or system32/SysWOW64 directory. GLEW has recently merged x86 and x64 versions, so both are now in single package.