It doesn't seem right, so even if it is technically grammatically correct it isn't idiomatic. Basically, you seem to be negating the eggs twice ("He likes not apples and not not eggs").
This may not be strictly accurate but it's the best way I can think to describe it. Doesn't is a contraction of does not; not is an adverb being applied to the does. As such, it negates everything that the does applies to, which in this case is the verb to like. An alternative (although archaic) way to phrase "He doesn't like X" would be "He likes X not". X in this case is "apples or eggs". This thus gives us "He doesn't like [apples or eggs]".
The way nor would usually be used is with neither (in the same way as German uses weder with noch). Both neither and nor are conjunctions which have no power to negate each other and both are intrinsically negative. Think of neither as either not and nor as or not. That being the case, "He likes neither apples nor eggs" could be rephrased as "He likes either [not apples] or [not eggs]." Each part negates only the noun (or noun phrase) it is applied to, not the entire idea of "liking".
As I said, I don't know if this is strictly grammatically sound, but it's what seems right as a native English speaker.
In British English at least it is more than widely used, it is the norm and is considered poor English is you write "neither ... or ..." That doesn't mean you won't hear people using "neither ... or ..." sometimes but then again you'll often hear phrases like "He don't know nuthink" :-)