Is my new puppy sick?

I don't think that your dog is sick. Like humans, dogs have their own individual characteristics. In my experience with the 5 dogs that we have, a new puppy (or even a dog) becomes timid as an indication of being homesick. It may be longing for its mother. With the hiccups, give it water. If it doesn't drink, use a syringe (with no needle) to force it to drink. The water will calm the hiccups.