The production that I worked on used carboard or a type of foam board cut into egg shapes, and propped up in the back with a "leg."  The tops were pre-cut with a zig-zag shape, and there were inverted Us at the bottom for "leg holes."  When the hatching took place, the actors rocked the eggs back and forth, poked their legs through the holes, and then pushed the tops off.  If you need more info, let me know, and I'll find out some specifics for you.

Let's see if I can communicate this without the use of hands, and from a year-old memory...

The Hatchlings were behind the eggs they hatch out of from the top of the show.  They were on the same level as Ugly's egg, but since it was huge, the mother could sit on a stool on a part of the stage that was higher, and appear to be sitting on the egg.  Ugly was also sitting behind his egg.  I've asked the director for some more information, so I'll get that to you soon.
Hope this helps!

Here is what my director sent back.  She gives two examples of contructing the eggs.  The way we did it was quite effective in our production.  It also made clearing the egg remains quite easy.

"We had a basic unit set which symbolized the swamp. We had various levels to it. We used foam board (flat surface) which we braced and the duckies hid behind them and as they were born we gradually saw various appendages until they stood up. In other productions, some folks construct the eggs out of styrofoam,sculpt them out where the actors can sit in them and then pop out."