The odd thing about this sentence is that it needlessly distances the child from the unfulfilled dream. That is, the unanswered wish is explicitly the child's, but the unfulfilled dream is simply "a dream unfulfilled"—and the use of and before "a dream unfulfilled" puts "a child's wish" and "a dream" on an equal footing, which further separates the child from the dream.
If the "a dream unfulfilled" element is there to serve as a quasi-poetical device for repeating the idea of "a wish unanswered" in different words (and I suspect that it is), you may be better off treating "a dream unfulfilled" as a restatement of "a wish unanswered." If you render the sentence as
Every empty box symbolizes a child's wish unanswered, a child's dream unfulfilled.
the repetition (in different words) of the "wish unanswered" idea gives that idea special emphasis without your being stuck with the problem of having introduced an unattributed dream in tandem with a child's wish. In place of "a wish and a dream," you get "a wish, in other words a dream."
If you move the adjectives unanswered and unfulfilled in front of the nouns wish and dream, respectively, the dream component's clunkiness becomes even more apparent:
Every empty box symbolizes a child's unanswered wish, and an unfulfilled dream.
Now the reader is even more conscious that the unfulfilled dream is hovering near the child but may not be the child's. To improve the situation, without running afoul of the relatively down-to-earth language being used, I recommend dropping the comma and the second "a" to yield a workmanlike description of a child who has both a wish and a dream:
Every empty box symbolizes a child's unanswered wish and unfulfilled dream.
Here, as in my earlier revision, the child's relation to both the wish and the dream is unmistakable.