Do you have the right to use a gun for hunting purposes?

In most states, yes. States have game laws that specify legal means of hunting- including guns. There are limits, such as which guns may be used for hunting what game, times and places, ages of hunters, licenses needed, etc. States may also have laws that forbid people from interring with a hunter that is complying with the law. You CAN lose your right to hunt with a gun (or by any means) if your break the laws.