The Christian Sabbath used to be observed not only with respect to Hunting, but ALL recreation and work. The puritans pulled this doctrine from the Fourth Commandment. Remember back years ago, in the 30s and 40s, when everything shut down on Sunday?
Historic protestant theology would argue that God is unchangeable and His commandments therefore are unchangeable so the Fourth Commandment is still binding (you can't murder people or steal, either - Thou Shalt Not Murder is still binding).
It is interesting to note that while many on the "conservative" side of the aisle scream bloody murder when the courts take the Ten Commandments out of the courts, schools and legal system, many turn around and throw the Ten Commandments out the window in their daily lives or when opening weekend arrives.
Of course, the puritans were definitely deer hunters and were proficient with firearms as they did live off the land much more closely than we do today. For them deer hunting was a necessity while for us it is a pasttime.
Instead of saying, Why can't I hunt on Sunday? The puritans would have asked, God, why do you give us a full six days each week to hunt, when what we really deserve is your wrath?