Probably one of the quickest ways to ship eggs in US is to use a small box. Most small lightweight boxes (2" x 2" x 3") in US can be shipped First Class Mail at an expense of under $1.50, with them reaching just about any destination within three days, not including Sundays of holidays.
If you have small boxes, I would recommend you do the following:
Remove the freshly deposited eggs from the surface on which they were deposited. Egg shells are usually fairly tough the morning after the night of their deposition. Eggs can also be left attached to paper surface.
The eggs can be picked of screen or paper surfaces just by using the fingers and fingernails.
Get a piece of toilet tissue or regular tissue and fold it so you have roughly a four inch square of at least two layers of tissue.
Count out the desired number of eggs into the center of the top layer of the tissue.
Fold the tissue layer in half so eggs rest in the middle.
Make two additional folds, one in from each side, so that the eggs are now surrounded by folds on three sides.
Make a final fold down so that eggs are now enclosed and bounded on all sides by folds or layers of tissue.
Use a piece of scotch tape so that the folded tissue will remain secure during shipping. Don't use so much tape that it will be difficult to
open the packet to get the eggs out.
Put the egg tissue foler in an envelope and fold the envelope so that the envelope fits snuggly inside the box.
Indicate species name and date of deposition on envelope.
Seal envelope in small box and securely tape on shipping label.
The use of tissue as explained above is probably not necessary. The eggs could simply be placed on paper in the envelope which you are going to put in a box.
A second method, and the one that I usually use is to get a piece of aquarium tubing and cut it into 1"; 1.5"; and 2" pieces.
I use 1 " for a dozen eggs; 1.5 inches for two dozen eggs; 2" inches for 3 dozen eggs. When many eggs are available, I usualy am quite generous in providing extras.
Take a small ball of kleenex tissue or bathroom tissue and stuff it into one end of the aquarium tubing.
Hold the tubing, open end up, over a small bowl (to catch errant eggs) and count out desired number of eggs into the tube. Stuff another ball of tissue into the
open end so that eggs are now secure inside the tubing. Tissue should be pushed in far enough that eggs will not be jostling.
I get a sharp pair of scissors and cut out a slot in a piece of fairly sturdy corrugated cardboard. Often I use a folded piece of corrugated cardboard and slot is cut through the fold, perpedicular to the fold so slot is actually cut through two pieces of cardboard. Stiff, plastic foam board also works well. The slot is just long enough and wide enough for a snug fit of the aquarium tubing.
I tape the egg filled tube into the slot, write the species name and date of deposition on the cardboard, and then place a small piece/layer of thin bubble pack (cut up an old bubble pack envelope if you have one) on one flat surface of the corrugated cardboard. Corrugated cardboard is about the thickness of a graham cracker with two flat paper layers and a series of folds between them.
Insert the pack into a regular letter sized preaddressed envelope and send First Class Mail.
Perhaps the diagram below will help.
Occasionally one of my shippers will get a "great idea" and try something different. One fellow was using small plastic cannisters to hold the eggs. He inserted each cannister into a bubble pack envelope and took them to post office. Most of the time the packing worked fine. But occasionally a postal worker would try to put the envelope through the mechanical processor which usually resulted in the cannister shooting out through one end of the envelope. Please use either the box technique or FLAT packing in an envelope with some protection for the eggs.