The best thing about hanging plastic Easter eggs is there's no single right way to do it. Hang them from pieces of ribbon taped to the insides of the egg, or tie yarn or twine through a hole in the top or bottom of the egg. Try several different methods to see which one you like best, based on your own custom crafting vision.
Flat Ribbon Hanger
Thin ribbon 1/4 to 1/2 inch wide -- especially if printed with a pastel pattern -- adds an extra decorating touch to your hanging egg decorations. Separate the top and bottom egg halves; then tape one end of a piece of ribbon inside the top half of the egg using cellophane tape. Fold the ribbon up and around the top of the egg to make a loop that ends at the other side of the eggshell top, like a hanger. Unroll more ribbon to make the loop as large as you'd like, depending on where you want to hang the egg. Snip the ribbon with scissors in the desired location; then fold the ribbon under the eggshell's rim exactly opposite the first taped edge. Tape the ribbon to complete the loop; then snap the bottom half of the egg back onto the top piece. If you prefer the egg to hang upside down, tape the ribbon to the bottom half instead. Add a small weight such as a pebble inside the bottom part of each shell if you're hanging the eggs outdoors to keep them from blowing around as much.
Some plastic eggs have holes at both the tip and bottom of each shell. Make a hanger for double-holed eggs by folding a piece of twine, jute or yarn in half, so it is long enough to hang from the desired location, plus at least 2 inches extra. Push the folded portion of the twine through the hole in the bottom of the egg. Tie a knot near the ends stick out through the bottom of the egg; the knot should be outside the bottom of the shell. Separate the egg halves to grab the folded twine and push it up and out through the top half. Use a toothpick if it is difficult to do otherwise. Snap the halves back together and pull the loop taut.
Add Your Own Hole
If you want the eggs to hang neatly from their tips or tails, but the eggs do not have holes in these areas, poke your own holes using a pushpin. Separate the egg halves and position the open end on a flat surface such as a table to poke the hole -- this greatly reduces the chances of the pin sliding and poking your skin.
While you could hang the eggs as-is from a tree or shrub outdoors, decorating the eggs first makes for far more fun. Paint the plastic eggs with chalk paint, designed to stick to plastic, selecting pastel colors such as robin's-egg blue. Add your own speckles by dipping a toothbrush in another color of paint and flicking your thumb across the bristles while holding the brush near the egg. Decoupage torn bits of newspaper or old damaged book pages onto the eggs for a little textual inspiration, or add stripes and designs to them with glitter nail polish.