Back in my gap-year, when I was travelling around Australia, I had an interesting encounter in the town of Broome. The Hostel I was staying at had a Bar downstairs, which was open to the public and frequently visited by the locals. I don't remember exactly how the conversation started, but one night I found myself talking to one of them. Even though the guy was friendly enough, he was kinda scary-looking: He weighed a good 120 kg and though he did have a proud belly, not much of those 120 kg was fat. He also had no hair and a bull neck, giving him a somewhat aggressive appearance. Most unsettling, though, was that he had apparently lost an eye. The odd thing was that while I was talking to him, I couldn't tell which of the two was the glass eye. So without me wanting to, my gaze kept flicking from one eye to the other, trying to figure out which one was looking back at me. Very distracting. It wasn't that his disability made me especially uncomfortable, it is just that I like to return people's gaze when they're talking to me, and looking a guy into an eye that is essentially not there is just like staring a non-disabled person at the ear when talking to them. It wasn't all that much of a big deal, I just felt it'd be more polite to return his gaze, rather than to stare at the side of his face incapable of returning mine.
At some point, we talked about Martial Arts. My visa was about to run out, and I mentioned that I considered going to Thailand next, to spend the rest of my time improving my kick-boxing skills - an idea that I had been toying with for some time. It turned out that he had done that in his twenties, and had brought back a wealth of stories from that experience that he didn't mind sharing. It was a very interesting conversation, but as I already said, his eye kept distracting me. At some point I interrupted him and simply asked him about it. I don't remember how exactly I phrased it, but I asked straight-forward, without much of a fuss, in a simple "sorry this keeps distracting me, would you mind just satisfying my curiosity so this is out of the way and we can move on?" -kind of way.
His reaction was somewhat unexpected: Not only did he not mind my question, he was positively delighted that I had asked. His face lit up, and before giving me an answer, he launched into a 10-minute-speech on how awesome it was that I had asked, how so many people were obviously confused but didn't "have the balls" to simply ask, and how great it would be if people were more open about their obvious insecurity instead of trying so desperately to pretend that the whole issue didn't exist.
Because physical disability is an issue. In this case, there was insecurity on my part, and I knew it showed. Asking a straight-forward question was the obvious way to deal with it, so I could focus on our conversation again. Which is exactly what we did. His eye wasn't mentioned again, I was more at ease, and although I usually am a rather nosy person, I didn't even ask how he'd lost it. Not because I didn't want to know or was afraid to ask, but simply because it didn't matter and our shared passion for Martial Arts was the more interesting topic at hand.
Pretending problems don't exist is something that never helps. And in cases where the Problem can be solved by asking one simple question in a polite manner, it is downright stupid not to do so. When you've never had any contact with disabled people before, your insecurity is perfectly normal and understandable, especially when you're faced with a particularly disfiguring disability. Trying to hide it does not work; nobody is that good an actor. What's more: Trying to hide it is dishonest. By trying to ignore the disability you imply that it is something that ought to be ignored, you're placing the whole issue under taboo, risking to trigger insecurity in the disabled person aswell. They will notice your insecurity anyhow, but what they won't know is whether your insecurity is rooted in disgust, pity, or simple lack of experience.
Simply acknowledging the issue and addressing it clears the table. Whatever they may think of your ignorance, they will at least know that your insecurity is not rooted in disgust or pity, plus you've demonstrated a genuine interest and willingness to be as understanding as possible.
In my (admittedly limited) personal experience this approach of a simple admission of ignorance plus demonstration of openness seldom (if ever) fails. The one thing you should make sure is to give the other a clear chance to say "I don't want to talk about it", in case you encounter somebody who, for whatever reason, really is touchy or just tired of the subject. Adding a simple preamble like "If you don't mind me asking..." should do the trick.
The same applies to other situations: It holds true for conversations with anybody who makes you feel insecure for whatever reason. Be it the disabled, gays, blacks, or the heavily tattooed. (Or Gingers. Gingers are scary.) It is not a shame to be discomforted by something you've never encountered before. Wariness of anything alien is evolutionary hard-wired into any living creature, for good reason. Being open-minded does not mean being perfectly free of prejudice - It means to recognize your own prejudice for what they are, and give anybody a genuine chance of proving them wrong. That is the very definition of our much-preached "tolerance" as it ought to be practiced.
Another thing I feel compelled to comment on: Staring. Most of us are told quite early in our lives that "it's rude to stare", which I don't think is true. I think what most people resent when being stared at is not the stare as such, but their own conception of what is going on in your head when you stare at them, the implications of that stare. Looking at a person isn't hurtful in itself - the implied pity or disgust such a look can convey however, is very much so.
So the next time you find yourself in a public place unable to turn your eyes away from a stranger's disfiguration, and that person suddenly moves to return your gaze, do not hurriedly look away.
Do not give them a look of pity, either.