) Slop the chickens is a regional (country southern) expression of mild expletive (others might be "well slap my face", "well I'll be darned").
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"Shucky darn" is, of course, built up of sound-alike euphemisms: "shucky" is a diminutive of "shucks", a euphemism for, um, excrement, and "darn" for ****. The times I've seen/heard it used are ironic or sarcastic, like the example---as if Garfield is just so upset that Jon's mother is pushing food on him (that he, Garfield, has a shot at getting some of).