A plane that creates such a cross section has to intersect exactly three faces of the cube. Let's consider three vertices connected with a common vertex by an edge.
A plane passing through these three vertices will create a trianglar cross section. Let's see it.
Each edge of this triangle is a diagonal of a face of the cube. Since the faces of a cube are congruent, their diagonals have the same length. This means that the edges of the triangle have the same lengths, so the cross section is an equilateral triangle. It is possible for the cross section of a cube to be an equilateral triangle.
