Good idea using your hand -- why didn't I think of that? I never paid attention to what my mother did... she probably used her hand!
In fact, I do use a square dish I happen to have but I also sort of scoot the egg over the top with the fork as opposed to trying to make it lay flat. This year the matza was really sort of curved, not flat at all, so no way I could have dipped it without it breaking.
On the other hand, if it breaks, and it almost always does, oh well. I just lay it in the pan with the pieces connected as they should be. It ends up holding together just fine. Especially the middle layers.