The first thing you would need to do is just set the egg carton in the sink and fill the cups with water.  Does the paper absorb the water?

If it does absorb the water fairly quickly, you could probably use them to reform into pots.  If they don't absorb, they are treated with wax or something similar, which prevents water absorption, which would make them unsuitable for your purposes.

If they do absorb water, just soak them in a bucket or tub until they are totally saturated (overnight may be best).  You may want to tear them apart first.  Once soaked, boil them in a large pot until they fall apart.  Drain them in a sieve or through a piece of window screen (formed into a cone). 

Then form them over some existing pots and let them stay there until they will hold their shape (they don't have to be perfect for plant pots), then remove and dry thoroughly.

I've never tried this, so I don't know how well the pots will form, or how well the plants will grow in them, or how well the plant roots will penetrate the paper pot.  Just keep in mind that the edges above the soil will act like a wick and cause the pots to dry out quickly.  (This was my problem with peat pots.)

Sue