According to this article from Wikipedia, the box-covering algorithm calculates the fractal dimension of a graph. The algorithm is based on the concept of distance between nodes; see for example the sentence:
A box consists of nodes separated by a distance $l < l_B$.
The distance between nodes can be defined also for directed graphs, so I think the algorithm should work also in that case. However, on the Internet, I cannot find any explicit statement about the possibility to use this algorithm for directed graphs.