you are right, the boolean did not work. You would probably need more triangles to do the boolean, the eye you have there is very low-resolution. If you switch to flat shading in meshmixer, or turn on the wireframe, you will see it is not many faces.
That is why the Remesh has "ripples" - it is a higher res mesh, but projected onto the low-res mesh you started with. The "smooth" version you see in the other images is not actually smooth (you can see the faceting in the first slice image, for example). It is just smooth shading, based on interpolating normals. So, the one with "ripples" would be a nearly identical shape when printed, even though it looks very different.
If you do want it smoother, you could run the Smooth tool on the remeshed version. I would suggest that you try to make it more spherical first. For example if you just select the back half, and a bit of the rim, and run the Smooth tool and make it very smooth, it will actually puff outwards on the back. This will be a better shape to work with.
You could also delete that eyeball entirely and put in a new sphere (it is just part of a sphere, from the looks of it). Regardless, you probably should shrink the socket to be as small as possible (or get rid of it entirely). You could use the sculpting brushes for that, or select most of it (make the selection brush large and click once in the middle, then right-drag until the orange comes out enough) and do an Erase/Fill. The issue is that the back wall is very thin, and it will be hard to get the boolean to not poke through.
You probably also need more triangles on the eyelid to get a nice-looking result from the boolean. Remesh it up to a higher resolution. Hopefully you will find that the boolean works in that case.