The cubic $x^3 - xy^2 - 2x^2 + x$ has critical points in $(1,0)$, $(0,-1)$, $(0,1)$ and $(1/3, 0)$. The determinant of the Hessian matrix is $-4(3x^2 + y^2 - 2x)$. It assumes the values $-4$, $-4$, $-4$ and $4/3$ in these four critical points. Thus the first three of them are saddle points.

The cubic $x^3 - xy^2 - 2x^2 + x$ has critical points in $(1,0)$, $(0,-1)$, $(0,1)$ and $(1/3, 0)$. The determinant of the Hessian matrix is $-4(3x^2 + y^2 - 2x)$. It assumes the values $-4$, $-4$, $-4$ and $4/3$ in these four critical points. Thus the first three of them are saddle points.

Added later: A simpler example is the cubic $xy(x+y-1)$ with saddle points in $(0,0)$, $(1,0)$, $(0,1)$ (arising from setting $\alpha=\beta=0$, $\gamma=1$ in the comments below).

The cubic $2x^3 + 8x^2y + 8xy^2 + 2y^3 - 3x^2 - 8xy - 3y^2$ has critical points in $(0,0)$, $(1,0)$, $(0,1)$ and $(7/15, 7/15)$. The determinant of the Hessian matrix is $-64x^2 - 112xy - 64y^2 + 88x + 88y - 28$. It assumes the values $-28$, $-4$, $-4$ and $28/15$ in these four critical points. Thus the first three of them are saddle points.

The cubic $x^3 - xy^2 - 2x^2 + x$ has critical points in $(1,0)$, $(0,-1)$, $(0,1)$ and $(1/3, 0)$. The determinant of the Hessian matrix is $-4(3x^2 + y^2 - 2x)$. It assumes the values $-4$, $-4$, $-4$ and $4/3$ in these four critical points. Thus the first three of them are saddle points.