Most commonly I hear the expression as “walking on eggshells.” That doesn’t really make sense, since if the eggs are already broken, you don’t have to be careful about walking on them. Occasionally I hear “walking on eggs,” but that doesn’t sound right for some reason.
Which do you say?
Walking on eggshells.
I think the point is that you don’t want to break them and eggshells are more fragile than whole eggs… which are surprisingly strong.
http://www.clichesite.com/content.asp?which=tip+817
There is no entry for “walking on eggs”.
I figured the reasoning behing ‘walking on eggshells’ was because the edges are sharp (and here I have a mental image somewhat akin to shell grit, which is definitely sharp) and so you’d have to tread carefully…
That’s just the conclusion I’d reached once upon a time. 
Seconding FilmGeek’s notion. Eggshells are very fragile and delicate, and one must step carefully to avoid crushing them (really, it’s a cliche, and not to be taken too literally). When someone approaches a subject as if “walking on eggshells”, it’s generally because the audience is also delicate and easily hurt/offended, and so the speaker must tread carefully.
I’d only heard the “eggshells” version, and understood it as FilmGeek explains it.
Also, to me it conveys some sense of the futility of your situation–if someone or something is so hypersensitive that it’s like “walking on eggshells,” it’s almost impossible not to upset it, just as it’s almost impossible to step on an eggshell, not matter how lightly, without breaking it.
If Mom says there’s eggs for breakfast, do you assume they’re still in the shell? Thus, the expression of “walking on eggshells” is quite specific. Otherwise, we might be walking on a scrambled mess! 