I've done research before, but I just went and checked it.
"Wild chickens" would be Red Junglefowl, scientific name Gallus gallus, that all of our domestic chickens are descended from.
They eat a wide variety of plants, insects, small animals, and similar stuff.
I looked several places, and NONE mentioned eating eggs at all.
Sites I looked at:
https://en.wikipedia.org/wiki/Red_junglefowl
http://animalia.bio/red-junglefowl
https://animaldiversity.org/accounts/Gallus_gallus/
https://eresources.nlb.gov.sg/infopedia/articles/SIP_541_2004-12-24.html
As a practical matter, making the egg takes more nutrients than the chicken can get by eating the egg. So it would be stupid for a chicken produce eggs, and then eat those eggs. Much more sensible to just not make eggs at all. (Which is what red junglefowl and other wild birds really do: no eggs except when it's time to raise babies.)