As many have noticed, you can't compete with the stores if you're selling the same thing. So it helps to sell them if your eggs are really not like store eggs; don't look like store eggs, and don't taste like store eggs. This is achieved by not using the same breeds that produce store eggs, and not feeding them the same feed as eaten by hens that lay store eggs. And to get started selling them, it helps to present them in a way that shows they're not store eggs - clear (recycled, as well as recyclable) plastic cartons work for me, to reveal the variety of pretty coloured eggs inside.
I sell my surplus to a small number of regular customers who are members of a CSA scheme, and we use the weekly veg pick-up boxes to distribute eggs (me to them) and payment (them to me); there may be something similar near you. My flock is mixed, so their eggs vary in size, shape, and colour, and my customers like that; next time you add to the flock, add a coloured egg layer. My customers understand that supply is erratic and seasonal (like the veg), and they put up with it. All but one pay in advance to ensure they get some as soon as the girls are laying, and the one that has been paying in arrears has finally twigged on why she is always last on the list to get any

They stay with me because they know that my hens' eggs really are different from store eggs, in taste and appearance and freshness, and we all think (tho' don't have the lab tests to prove it), that they're more nutritious because they're laid by hens that don't eat homogenized pellets but recognizable foods and they forage all day long.
I don't charge more for them than I need to cover my costs; I am selling surplus eggs from a hobby flock, it's not a business; the most I've supplied in any one week is about 3 dozen. I don't think this model would work if I was trying to shift hundreds of eggs a week, and it definitely wouldn't work if my eggs were indistinguishable from store eggs.