You are confusing the square of the spin ($\vec{S}^2$) with its 3 components ($S_x, S_y, S_z$).

The square of the spin $\vec{S}^2$ has only one eigenvalue. It is $s(s+1)\hbar^2$, which is positive. The electron has $s=\frac 12$ (no negative sign).

The components $S_x$, $S_y$ and $S_z$ each have two eigenvalues (a positive and a negative). These are $+\frac 12 \hbar$ and $-\frac 12 \hbar$.

You are confusing the square of the spin vector ($\vec{S}^2$) with its 3 vector components ($S_x, S_y, S_z$).

The square of the spin $\vec{S}^2$ always has the value $s(s+1)\hbar^2$, which is positive. And the electron has $s=\frac 12$ (no negative sign).

The components $S_x$, $S_y$ and $S_z$ each have two possible values (a positive and a negative). These are $+\frac 12 \hbar$ and $-\frac 12 \hbar$.

You are confusing the square of the spin ($\vec{S}^2$) with its 3 components ($S_x, S_y, S_z$).

The square of the spin $\vec{S}^2$ has only one eigenvalue. It is $s(s+1)\hbar^2$, which is positive.

The components $S_x$, $S_y$ and $S_z$ each have two eigenvalues (a positive and a negative). These are $+\frac 12 \hbar$ and $-\frac 12 \hbar$.

You are confusing the square of the spin ($\vec{S}^2$) with its 3 components ($S_x, S_y, S_z$).

The square of the spin $\vec{S}^2$ has only one eigenvalue. It is $s(s+1)\hbar^2$, which is positive. The electron has $s=\frac 12$ (no negative sign).

The components $S_x$, $S_y$ and $S_z$ each have two eigenvalues (a positive and a negative). These are $+\frac 12 \hbar$ and $-\frac 12 \hbar$.