Solution:

Let one vertex be $A=(0,0)$, and let the adjacent vertex with y-coordinate $1$ be $B=(x,1)$ with $x>0$.

Rotating the side vector $(x,1)$ by $90^\circ$ gives the next side vector $(-1,x)$, so another vertex has y-coordinate $x$. The remaining y-coordinates are $4$ and $5$, hence $x=4$.

The side length squared is $AB^2=x^2+1^2=4^2+1=17$, which is the area of the square.

Thus, B is the correct answer.