Coconuts can be found throughout the Shore and sometimes in the other biomes, and most easily found and cooked at the shore where you start. Throw them at a surface to break them into edible halves, and cook before or after breaking them. The remaining food items can be cooked at the campfire at the end of the biome or with a portable stove.
Beehives are found in the Tropics, surrounded by buzzing bees. You can either grab the hive and then run away from the bees, which will poison you a lot if they reach you, but will eventually stop chasing (if you drop the hive they will give up faster); or you can throw an item to knock down the beehive and the bees will disappear. Throw the beehive at a surface to break it into edible honeycombs. In the Roots biome, honeycombs can be found within hollow tree stumps.
Yellow winterberries can occasionally be found in trees in the Alpine biome, but will always be found on the head of a capybara in the capybara pool which spawns in both the Alpine and Mesa biomes. In the Alpine biome, it is a hot spring that spawns somewhere on the mountainside. In the Mesa biome, it is an oasis with coconut trees that spawns somewhere in the first area where tornadoes roam.
Big eggs are found in the Caldera biome, in bird nests scattered around the rock pillars. Throw them at a surface to get the edible egg which can be cooked before or after breaking it (though rarely, you get a cooked chicken instead, which will not count). It may also be cooked if you are engulfed in lava while holding it, but don't let it get burnt.
Once you've eaten all four cooked foods, you just have to escape the island alive.