I've heard ' button up your shirt or button up your coat '. But I'm not sure If we could say 'button your button' or 'put on your button' as well. Let's wait for somebody else to reply.
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TomJif it's wrong to say 'button up your shirt' or 'button up your coat'No, it's not wrong. Those sound completely normal, as do the ones without "up". "up" adds the feeling of "all the way", "all the buttons" so that you end up nice and neat and cozy, all "buttoned up".