UPDATE: THIS IS FALSE.
I came up with this question some day and have been working on it for some time:
Every bipartite graph $G$ is an induced subgraph of a hypercube graph $Q_n$.
Here the hypercube graph is a graph defined as follows:
The hypercube graph $Q_n(V_n,E_n)$ with $V_n=\{\text{all subsets of }\{1, 2, … , n\}\}$, $E_n=\{\{u,v\}:|u\triangle v|=1\}$, where $A\triangle B=(A\setminus B)\cup(B\setminus A)$.
With some small examples I believe this is correct.
Here's what I've trying to do: by block decomposition we reduce to the biconnected scenario. Every biconnected graph can be constructed as follows: starting from a cycle, and adding paths to the graph. If we can present a way to add the paths in such a way that the add path itself is longer than the original distance of the two ends (in the graph before adding this path), we can always label a subset of $\mathbb{Z}_+$ on every vertices added such that two vertices are incident if and only if the subset labelled on them has the difference of exactly one element (i.e. $|A\triangle B|=1$), because every cycle is even. (This should be true, I think) In this way we can embed this graph to some $Q_n$.
Now it remains to show the "if" is true. I tried to add the shortest path every time but this doesn't seem to work.
Best Answer
$K_{3,3}$ is a counterexample, thanks @radekzak. If a vertex is labelled $\varnothing$ then all its neighbours are singletons, and there's no appropriate choice for their another common neighbour. $\square$