When I help a chick hatch and it has bits of shell or that gunk on it, I usually soak it a cup in warm water, right around 100 degrees and remove what I can. You don't want to chill the chick so use warm water. I especially want to get he shell off. Leaving some of the gunk (which will plaster the down to its body) isn't horrible as long as it doesn't prevent the chick from being able to move. It should wear off within a week. I've never had other chicks peck at the gunked down, that would worry me about leaving bits of egg shell.
One time I gave a chick like that to a broody hen, along with several other chicks. Egg shell was removed but the down was plastered down. The weather was pretty warm so the messed up down did not cause it to chill and the hen accepted it fine. Within a week the down had fluffed up (dust bathing probably helped) and I could not tell it from her other chicks.