You’ve had a lot of good responses about rabbits so I won’t add to that particular topic. However, as a chicken (and duck and goose) person, I thought I’d share some thoughts.
To start with, chickens without supplemental feed won’t produce eggs as well. Prior to widespread grain feeding and selection for production, it was normal for hens to lay 60-80 eggs a year. Eating chicken was also not a frequent thing back then - only for special occasions or when doing seasonal culls. This is just something to keep in mind and isn’t a “good” or “bad” thing.
Chickens can glean about 30% max of their required nutrients from the non-seed portions of plants. They require a calorie-dense diet. In the wild, this is largely insects, plant seeds, and any meat they can scavenge. Truly feral populations are largely limited to semi-tropical and tropical locations where insects and plants are present year-round.
In the mid-19th century, people kind of went bonkers about breeding animals and plants and importing new types. This is the time when many breed associations were created and when many breeds were formally made a “breed”. A number of chickens were imported from Asia (primarily China) and they were used to develop new breeds. These breeds all have a common feature from the Asiatic breeds that didn’t exist in European and Near East breeds: brown eggs.
Nowadays, people call these relatively modern, heavy(ish)-bodied brown layers “heritage” breeds - Plymouth Rocks, Orpingtons, Sussex, etc. They were developed during the time period when grain became cheap enough to feed chickens. These breeds are a bit like a plant that was bred for high-input farming.
Prior to that, grain was food for humans and high-value draft/riding animals (horses, cattle). Chickens were usually just “underfoot” and would clean up scraps from the more valuable animals and whatever the humans tossed their way. In countries like England where hedgerows were common, chickens could glean a lot of their needs from foraging there. We also have references to so-called “dunghill chickens” which were mostly game chickens that weren’t “game” enough for fighting and lived semi-feral on the dunghills (compost piles).
When we look at chickens that go feral and at pre-grain chicken breeds we see that they are mostly small birds. Two exceptions off the top of my head are the Dorking (modern weight around 8#, first referenced in 43 AD) and the White Faced Black Spanish (modern weight around 8#, pre-1600s). Otherwise, they tend to be much smaller, flighty, and game-like. Breeds like Leghorns, Hamburgs, Anconas, Andalusians, Campines, and various Game breeds - largely (and very sadly) out of favor amongst modern chicken owners.
On a farm setting, the most self-sufficient breeds you could choose would be the smaller birds that lay white or tinted eggs. Opt for breeds that are described as flighty - they’re likely wickedly smart (for a chicken) and most likely to survive while free-ranging. If they have access to be around larger animals they’ll love cleaning up spilled feed, scratching in their poop, and the presence of the larger animals deters some predators like hawks.
On a town/suburban lot setting, I’d opt for bantams. They need less feed input and can be quite productive for their size depending on breed. Choosing a feather-legged breed will reduce their scratching (but also means less food foraged).
