There is no simple answer to this question as it depends on a number of factors, including the type of 556 you are using, the game you are hunting, and the state or country in which you are hunting. In general, however, it is legal to hunt with a 556 rifle if you have the proper license and permits.
If you are using a 556 for deer hunting, for example, there are many states in which this is perfectly legal. In some states, however, there may be restrictions on the type of 556 rifle that can be used for deer hunting. For instance, in Pennsylvania only certain types of 556 rifles (such as those with a minimum barrel length of 16 inches) can be used for deer hunting. Additionally, most states have specific regulations regarding the use of firearms during deer season- so it is always best to check with your local wildlife agency before heading out on a hunt.
In terms of other game animals, the rules vary depending on the species being hunted and again depend on where you will be doing the hunting. In many cases it is perfectly legal to hunt smaller game animals such as squirrels or rabbits with a 556 rifle. However, there may be restrictions in place regarding the use of certain types of ammunition- so it is always best to check local laws and regulations before heading out into the field.