We can even assume the lattice is triangular: for each hexagon, add a point in its center and connect it to the six vertices of the hexagon. The conclusion holds for this larger lattice as well: Assume WLOG that we can find points $A,B$ belonging to this lattice s.t. if $O$ is the origin of the plane, $OB$ is obtained from $OA$ through a rotation of angle $\frac\pi 2$ around $O$. This is equivalent to finding integers $a,b,c,d$ s.t. $\frac{a+b\varepsilon}{c+d\varepsilon}=i$, where $\varepsilon$ is a solution to $x^2+x+1=0$ ($a+b\varepsilon$ represents the point $A$, while $c+d\varepsilon$ - the point $B$). In turn, this implies that we can find rationals $u,v$ with $u+v\varepsilon=i$. This is claerly absurd, since eliminating $u$ between $u+v\varepsilon=i,u+v\bar\varepsilon=-i$ gives an irrational value for $v$.