$n$ vectors are in $\mathbb{R}^{n-1}$ with dot product $-1$. Does this force that they are in $\{-1,+1\}^{n-1}$?
Certainly not. Take the vertices of an equilateral triangle centered at the origin in $\Bbb R^2$. The vectors will form angles of $2\pi/3$; if you make their lengths $\sqrt2$, the dot products will be $-1$. Specifically, take $$v_1 = \sqrt2\big(1,0\big), \quad v_2 = \sqrt2\big({-}\frac12,\frac{\sqrt3}2\big),\quad v_3 = \sqrt2\big({-}\frac12,{-}\frac{\sqrt3}2\big).$$