Wow, to be perfectly honest, I'm not sure what he's doing!? The chick looks ok.... in the video it looks like he has hiccups or something. I don't think he's ill..... Is he the only one peeping? I hear the noise, but can't tell where it comes from...
I assume you're feeding chick starter?
It sounds like the chick is ok, a little sneeze now and then is ok, my adults do it too....freaked me out the first time I heard that high pitched squeak! - Just keep an eye on it, make sure it gets plenty to eat.... if he's running around - sounds healthy to me!
I'm sorry to hear she rejected them. I think I would take that one out before they all get sick - if you really feel the chick is sick....
It's sneezing? Are the nostrils runny or having a discharge?
It may just be that the chick is eating and swallowing too quickly. Does it appear sick or droopy looking? If it's eating, drinking and pooping normally - maybe you're overly concerned. I was, the first time I had chicks it was very stressful to me!