No you can't, but you can use a lead to make them follow you.
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no
You can't.
no
No. You can suggest it on the Minecraft Forums and someone might make a mod of it.
You can not "own" a chicken in minecraft. If you hold wheat in your hand the chicken will follow you but it will never belong to you. You could always build a pen to lock chickens in. From there you can help them reproduce by feeding them wheat.