Is De Bruijn Graph Hyper-cube? I do not know if I miss something in the definition of a hyper-cube, but as far as I understand, Hyper-cube graphs have 2^n vertices and if written in binary form, "one-bit difference" strings of numbers are linked with an edge. 

From this definition, De Bruijn graph does not look like it is a hyper-cube, because 000 and 010 for example, are not adjacent.
No. I was writing an answer based on vertex and edge counts, but then I realised that there's a simpler argument: the de Bruijn graph of strings of length $n$ over an alphabet of $m$ symbols has $m$ self-loops, whereas a hypercube has none.