Yes, this is true. Thanks to Sukhada Fadnavis and Seva for pointing out in the comments that the argument I had written here was wrong. Instead I will point you to the paper where this is proved

"Bulky subgraphs of the hypercube", by Andrei Kotlov, Europ. J. Combinatorics (2000) **21**, 503-507

As far as I can tell from looking at the literature, it is not known if there are configurations of more than $2^{n-1}$ vertices for which one can not find $n+1$ of them which induce a tree with an edge in every direction. This would be a strengthening of the result in question.