Some of the items in Earthbound are just joke items. I don't remember if it's the Ruler or the Protractor, but one of those can sometimes cause your character to go berserk and do a "SmaaaaAAAAAAAAAASH" attack.

You don't sell the mushrooms to the guy who has one on his head. You sell them to the man with the purple hair who hangs around in the hospitals and claims to be a healer.

In Peaceful Rest Valley, you can also sell them to that girl near the entrance who says she's looking for them.

Having a chicken in your inventory prevents homesickness, but if you try to use it, it'll run away. It's best to sell it if your inventory's getting cluttered.