The book is correct: it should be vegetables (plural).
When it's a count noun, and it's a general statement, then the plural form is used:
✔ I don't like cars.
✘ I don't like car.
✔ I don't like movies.
✘ I don't like movie.
When it's a mass noun, then the singular form is used:
✔ I don't like bacon.
✘ I don't like bacons.
✔ I don't like gravel.
✘ I don't like gravels.
Problems occur when something can be considered either a count noun or a mass noun.
✔ I don't like cake.
？ I don't like cakes.
The first sentence is fine—it's using cake as a mass noun. The second sentence is also fine—although it's not as common (and therefore looks a little strange) because we don't normally refer to the count version of cake in this fashion without using some kind of adjective.
Strangely, when we use an adjective, the mass-noun version becomes unidiomatic:
✔ I don't like small cakes.
✘ I don't like small cake.
However, I don't like cakes without an adjective, while uncommon, is still acceptable.
As with cake and cakes, fruit has the same issue. The count form of fruit is fruits.
Therefore, we have this:
✔ I don't like fruit.
？ I don't like fruits.
As before, the first sentence uses the mass-noun sense of fruit. The second sentence uses the count-noun sense of fruits. While fruits is is not as common, it's still accepted.
But vegetable has no mass-noun sense—it's a word that's always used as a count noun:
✔ I don't like vegetables.
✘ I don't like vegetable.
Since we don't use vegetable as a mass noun, it's singular use here is wrong.
Note that I can't say why we use some words as mass nouns and not others. It's simply the case that our use of the language has worked out that way.
So, returning to the original sentence, fruit can be either a mass noun or a count noun, but vegetable can only be a count noun.
Therefore, there are only two correct versions of the sentence:
✔ I like neither fruit nor vegetables.
✔ I like neither fruits nor vegetables.
Although fruits is less common, in this particular case my personal choice would be to use it in order to maintain parallelism. However, both are still correct—and it's a matter of style which you want to use.
The confusion over vegetable as a mass noun may come from constructions such as the lyrics to the "Major-General's Song" from Gilbert and Sullivan's The Pirates of Penzance:
I am the very model of a modern Major-General,
I've information vegetable, animal, and mineral,
I know the kings of England, and I quote the fights historical
From Marathon to Waterloo, in order categorical;
But, as with most songs, the syntax here is poetical and nonstandard. It only appears as if vegetable is being used as a mass noun. In fact, it's being used as an adjective:
I have information that is of a vegetable, animal, and mineral nature,
Then there are things like the crossword clue:
neither vegetable nor mineral
(The answer to this is commonly "animal.")
But that clue is missing articles, in the same way that headlines frequently omit articles and other components of grammar.
In reality, the full version of the crossword clue should be:
What is a word for something that is neither a vegetable nor a mineral?