Personally, I think it would be more trouble than it's worth.  Making the cup parts would be easy enough with a plaster mold, but the strength comes from the overall design (edges and attached lid).  Remove the lid from an existing egg carton and it becomes far more fragile.

But if you more time on your hands than you know what to do with...

1.  Find a rigid container that is about 25mm larger in all directions than a whole egg carton with the lid opened.  A double-walled corrugated box might be okay.  You will have to rip the box apart to get the casting out, so don't use metal or something indestructible.  Fill the box with sand or dry rice or beans to the height you will be filling it for the casting.  Measure the dry material with a measuring cup to see how much plaster you will need to mix.  Write it down!

2.  Melt some wax (old candles would do) over VERY LOW heat or in a double boiler (use a thrift shop container - you can't reuse it for food).  THIS IS A FIRE HAZARD!  KEEP YOUR ATTENTION ON WHAT YOU'RE DOING!  Then use a paintbrush to paint the liquid wax all over the open carton, inside and out.  Also paint the inside of your cardboard container with the wax.

3.  Place the waxed carton in the bottom of the box upside down and seal the edges with more melted wax.

4. Pour mixed plaster slowly over the carton, probing with a stick to make sure all gaps are filled.  Let set undisturbed.  The setting of plaster is a chemical action, not a drying action.  Let it set until the mold is room temperature.  Usually, if it is cool, it is still damp.  Tear away the outer box, roll the mold over, and remove the carton carefully.  New plaster is somewhat fragile, so take your time.  Once the mold is freed from all the cardboard, let it dry more.

5.  Mix up your pulp and apply to the mold.  Plaster will absorb moisture and help the papier mache to dry faster.

OR

Put an ad on Freecycle for clean egg cartons.

Sue