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Actually, by searching on mathscinet I found Daverman, Robert J. Every crumpled n-cube is a closed n-cell-complement. Michigan Math. J. 24 (1977), no. 2, 225–241. which proves the same result for dimensions at least $5$ and Daverman, Robert J. Each crumpled 4-cube is a closed 4-cell-complement. Topology Appl. 26 (1987), no. 2, 107–113. which proves it in dimension $4$. So it always holds. I'd be interested in an easier proof!
@IanAgol : The theorem of Lininger referred to in the wikipedia article is very cool! You should post an answer with that reference so I can accept it (though I guess it only gives the $3$-dimensional case; my suspicion is that no one will respond with a more general result).
