I believe Nebraska, Illinois, New Jersey, and Hawaii are the four states that still don't allow private citizens to carry firearms for personal protection.
The article above says Nebraska allows concealed carry, but has an "opt-out" option for landowners to ban handguns. That would be similar to Texas' statutes; you can post a Section 30.06 sign and a CHL who carries on your property risks loss of license.
Now, open carry may be regulated; even in Texas, one of the more gun-liberal states, open display of a weapon in a public area is specifically defined as disorderly conduct. I haven't seen a map of open carry states but I would assume open carry is illegal in Nebraska as well.
Of course, this may be rather moot; Imagine a local mall, packed with Christmas shoppers, and someone goes postal. Remember Rule 3: Always be sure of your target and what is around and behind it. Even if you're a crack shot with your CCW, you'll have a lot of people scrambling around in front of, around, and behind the gunman trying to get out of the way, virtually guaranteeing you will not get a clean shot at the assailant from more than arm's length until said assailant has had the opportunity to fire a lot of bullets into a lot of people. Wouldn't the Brady Bunch have a field day with the small detail that a bullet fired by a CHL holder while taking down the assailant missed or overpenetrated and struck a bystander? The bullet wouldn't even have to draw blood; it could graze, or overpenetrate and bounce off, and the newspapers could imply serious injury by simply mentioning "a bystander was hit by a bullet fired in taking down the gunman", and the Brady Bunch would tell anybody who listens that it proves their point; "amateurs" wielding guns are just as dangerous as the criminals.
And you would have to admit, if it were anyone but a CHL, who is required to qualify with the type (and often caliber) of weapon they carry, the Brady bunch would have a point; a gun in the hands of someone who is not proficient with it is indeed just as dangerous as a criminal gunman firing randomly.