Fun! Think third one works best as main logo, but it would be nice to have the chicken picture used elsewhere as a secondary mark. Its tail looks like its foot?

I would go with Number one.
The plectrum-bodied chicken image is strong, original (AFAIK), and cute, and the illustration style and lettering go well together, and suit the name.
I actually don't mind the way you've interpreted Duffy Script.
But the Initial C needs to be bigger, so that it doesn't look like ducky's, and rotated—C's that have the bottom terminal stronger than the top look too much like G's. And the baseline needn't be so firmly indicated; sure, for stability it may be more apparent than the median (top of the x-height), but as the median is bounced, then it is wrong to have the baseline dead straight. Similarly, the height of the ascenders could be slightly different, just as the shape of their loop is different.

Number two, with two ideas, dilutes the branding.

Number three is a bit thin and weak, especially in light colour.
The claw-as-apostrophe is a poor metaphor—it's not the right shape for an apostrophe and the claw mark is not particularly chicken-ish, so the concept is forced and too obviously clever.

Hi Mr. Shinn! It's an honor to get a critique from you!! Definitely inspired by your wonderful creation! I hope you didn't mind me modifying it a bit for this project. :)

The design is still currently in the works.. Everyone's feedback has been very helpful in the revision process. I'll be going with a different font, though. Someone has pointed out that it looked similar to the logo of another local restaurant, so sadly.. i guess that's the end of that.. Does anyone have any suggestions?

Thanks!!!