Why are some of these vertices being ignored by edge loop selection? I have a mesh which has quad vertices but when I try to click to select the edges of the mesh, it stops where the red circles are. But as you can see around the edge of the mesh, the rest is selected. Shouldn't it be able to select the edges since its all quad faces, there are no Ngons or triangles.
Your topology has corners, which break the edge loop.   

You can solve it by a small Inset, to keep your edge loop connected.



If you want to continue selection through corners, just click again :).