Yeah, definitely not at all possible. For one, how would two separate species get in the same egg? Even if a hybrid was possible, it wouldn't be two separate species in the same egg, it would be one baby that was a cross of the two species.
So either this was made up as a hoax, or, since the OP mentioned they were tossing out multiple eggs, one was a chicken egg and one was a duck egg and they landed near each other and so OP thought they came out of the same egg.
There is absolutely no way at all that both of those were in the same egg.
Found this interesting web page. It's huge, just browsed it. But there seem to have been other "Chuck" sightings.... (called...Chuck. Who knew!)
http://www.macroevolution.net/chicken-duck-hybrids.html
ETA: There is an actual, narrated Chuck video!!! Whoa. It looks pretty real.
That picture of a 'chicken/duck hybrid' at the beginning is just a duck that's had the webbing cut out of its feet. Blow up the picture and you can even see remnants of the webbing still attached to its toes. I feel really bad for that duck because it was likely mutilated to promote a hoax.
That second one is either a very sick chicken that's walking upright because it has something wrong with it, like egg binding, or it just walks that way. All chickens float. Chickens can swim. Just because it can float doesn't make it half duck.
You'll also note that as soon as they wanted to do testing on her, suddenly her remains were no longer available and had actually been burned, not buried, and the remaining feathers that they had had been 'thrown away'.
This is just biologically impossible. These two birds aren't just different species, they're different orders entirely. That would be like saying an elephant could breed with a horse because they're both mammals.