Hiccups in puupys are very normal, cause no damage, and nothing to worry about. What is happening is that your puppy, and puppies in general, by nature have had to struggle for food with siblings, and as a result they eat or drink to fast. When they do, they intake a large amount of air into their stomach, casuing them to hiccup.
What you might try is to get them to slow down, even if you have to pick up their food or water, place it back down, and repeat this until they slow down.
Another thing you can do, and I still do this with the dog we rescued, is to stroke them gently while they eat. This helps to relax them and at the same time bonds you together. My little guy was thrown out to die by someone as a puppy, and for the first couple of months he had severe hiccups. This cured him, but it took some time.