DirectX 11 client download link.

[Sylve]

I thought it would just be a check box like what WoW has, but they keep saying it's a completely different client which leads you to believe that the full game will need to be downloaded for it.

When you check that box, you need to restart the entire client in WoW.
The DX9 and DX11 there are actually two different clients, it just loads the appropriate one to the state of the checkbox.
The DX11 in WoW came with a patch, So we never really need to download it as a separate thing.
I'm fairly sure the 3.0 patch that gives us the Expansion content will include the Dx11 client.