Disclaimer: I am a strict-constructionist 2nd Amendment guy, so I believe that the phrase "the people" in the Second Amendment means "every individual," just like it does everywhere else in the Constitution. (And the Supreme Court agrees with me.)
However, I will say that I have very little confidence in the judgment and gun-handling skills of the vast majority of Americans, and the thought of running into one armed in the back-country chills my spine. Perhaps I could distract him by tossing a microwaveable burrito into the underbrush and running the other way.
On the other hand, I can think of precisely (only) one good reason to carry a firearm on a hiking trip in North America: Bears. And even that is a stretch. 99.9999999% of bear encounters are non-threatening, I know. That said, if I run into bear number 0.00000001 I want to know that I can take him with me.
I was recently planning an Alaskan trip and was very uneasy about not being able to carry a rifle into a national park. I know that Alaskan natives are not shy about carrying firearms in bear country. Bear spray is good for morale, but discounting individual testaments when one looks at the well-constructed studies no one is really sure how well it works. At all. Rarely, some bears are even attracted to the smell. The Alaskan, B.C., and Yukon natives I talked to seemed to think bear spray is charming, but not very practical.
So I changed my trip to one where I could carry my rifle, even though I had to jump through hoops to do it (because part of the trip was in Canada). Never did see a bear. But that's the thing about guns: You'll probably never need it, but if you do you'll need it VERY BADLY.
But back-country Alaska is a special circumstance, isn't it? As I said, I'm a 2nd Amendment guy, but from an environmental viewpoint I just don't trust most beer-swilling Americans. Thus I also have worries about some camper-cowboys gunning down elk "that were about to charge." Honestly, how can we expect an urbane retired New Yorker who has barely ever left the city to make informed judgments about when an elk constitutes a threat? There are too many people in many (not all) of our national parks as it is, dropping litter and stomping on the undergrowth, and I really don't want those teeming masses armed. They have driven me into the back-country already.
I admit I'm being elitist, here, in that I have no objections to ME going armed. I know what I'm doing. (Seriously.)
Anyway, wouldn't the people who really need a gun in a national park probably be those back-country campers, not the drones there to see Old Faithful? I haven't heard of many bear maulings in the vicinity of Old Faithful. The tourists there are all either in the lodge or camped in massively populated campgrounds. Perhaps a back-country pass could act as a gun permit as well? I think that would be ideal.
Rats, I'm getting really conflicted now, because I just remembered some ruminations of mine about an Appalachian Trail hike. There have been several well-publicized murders on the trail (rare though they are). AT through-hikers make excellent targets because they can't carry a gun for self-defense, since they travel through several areas where that is illegal. I'm not a big guy, so I'm pretty certain that if I ever through-hiked the AT I'd commit at least one felony, because I wouldn't do it without a derringer (American Derringer Co. "Lightweight", 7.5oz not counting ammunition). The most dangerous animal on Earth is man, after all.