I'm afraid the way you are doing it will yield zero results.
You need to get some seed starting potting soil and some plastic dixie cups or small pots with drain holes. Start your seeds inside on a warm window sill or a place that stays warm (to keep the soil warm and for seed germination..this is crucial).
Put saran wrap over the top (after watering) and put a rubber band around the top to hold wrap in place.
Keep the soil moist. You will see water bead on the wrap..this is good.
Your seeds should germinate in a few weeks, maybe longer.
Be patient. Once you see the first two leaves, take the wrap off. Make sure the soil is moist and watch your plants grow.
Now, look up "hardening off pepper plants".
You will need to do this before you actually plant them outside.
This happens when they have at least 4 true leaves and about 3-4" tall.
Good luck!
Dave

Welcome, April - you've come to the right place. We'll have you up and running soon.

Your egg carton setup is not optimal, but it won't affect germination so let's start with that first.

It may not be freezing where you are, but peppers want WARM temps to germinate. (Also moisture but not wet.) Bring them inside and put them someplace warm - 80F if you can find a spot (in your oven with the light on is one possibility), otherwise warm room temp. Then we'll see if you get some sprouts.

Where did you get the seeds?

BTW, why the egg shells? Are you thinking that they will provide nutrients? Are there holes in the bottom of the shells?

Dennis