Sure, people might make fun of you, but y'know what? Accept it. Your eyes are your lively hood and you don't want to lose them.
In many cases, you won't need them... but if you're cutting metal, snipping parts that have the tendency to fly across the room or working with power tools (like a Dremmel) then it's a good idea to have a pair handy - AND USE IT.
The bonus points you'd get for looking like a pirate would not make up for losing an eye.
I searched a bunch online, but couldn't find any examples of broken cut off wheels. (rather surprising)
Damage like this can happen very easily and is a serious safety risk. I kept these after chopping up some larger steel equipment for disposal as an example of what can go wrong. Usually these wheels wear down, but sometimes you'll angle them wrong or they'll catch on a bur and instead will shatter and explode. When hobbying, we're usually not using any equipment on this scale, but a smaller dremmel's cut-off wheel works the same way and can be just an dangerous to our eyes as the larger versions. Also, if I can cut a piece of wire that shoots off with enough force to shatter the glass of my fluorescent light, what do you think it would do to your eyes?
These and some dremmels cut-off wheels are fibreglass reinforced; that's what helps to hold the wheel together. Not all wheels are reinforced like this, and look here how sometimes that doesn't help.