@Count Zero:
"Modern" flat-Earthers believe the sun and moon both remain on a plane roughly 3000 miles above the earth's surface. My argument states that the earth couldn't possible ever get between them. Not to mention that their model of the earth is about 260 times the diameter of the sun and moon.
@Gazpar:
I haven't run into this "shadow object" business before, but it figures that someone would come up with that, sooner or later. He decided to make it 5 to 10 miles in diameter and close to the sun, I guess so he could claim that's why we can't see it. But that doesn't work, either. At only 1/6 to 1/3 the diameter of the sun, nothing even close to a total lunar eclipse could ever occur. A penumbral eclipse is all we would see, and even that would only be a slight darkening of the moon. Let's remember that they say the moon and sun are about the same size, so this object would also only be 1/6 to 1/3 the diameter of the moon. So it could never cast an umbral shadow which covers the whole moon, no matter where you place it.