Some of you are confusing apples and oranges.
National Forest: Perfectly permissible. Unless there is some local rule, but that's not the general case. For example, during fire season access and what you may or may not do, may be restricted. Also, many National Forests have designated camp grounds. Usually, shooting within the camp ground itself is prohibited, but they will be posted. You may even carry and shoot real steel guns in National Forests.
National Park: The carry of real steel guns is strictly prohibited. So, I would presume Airsoft is also.
State Parks: It depends on the State.
I think you would be taking a risk. First of all, it's government property. If they decide to prosecute under any law, they don't have to ask if the land owner would be willing to go along with it. Which means if they want to prosecute for systematic littering with BBs, they can. I'm sure US attorneys don't have time to mess with kids with airsoft guns, but what if they decide to make a point? You don't want to mess with government lawyers. Who knows what other silly laws they'll find.
Secondly, hunters is one thing, but people with what looks like military equipment is entirely different. People don't call cops on hunters in national forests, but they would if they see a band of militia, practicing war games. Of course, since it's miles away, that risk is minimized, but still, all it takes is one person with a cell phone.
You may not have know that somebody was near by, but unbeknownst to you, there was a person, and you pointed your gun at him. Pretty much in all laws, that's an assault. Battery is actual contact, assault is often defined as "perceived threat," in many jurisdictions. It has nothing to do with your intention nor your capability. If a reasonable person could "perceive" that he/she was threatened, you are in trouble. Which is why it's not a good idea to play in a field where other people could just wander in. Most people wouldn't press charged against teens, but what if somebody really wants to make a point? Even if government wants to let it slide, if the "victim" insists, they have to go along with it. Far fetched? Yes, the chances of that happening is small. But do you really want to risk it?