A square in a curve - Colegiul Ortodox "Mitropolitul Nicolae Colan" (Cluj-Roumanie)

We draw a closed curve on the plane. Can we always choose 4 distinct points on this curve, A, B, C, D, such that ABCD forms a square?
To begin, one can study specific curves, for instance, those formed by segments (polygons) and/or having certain symmetries. One can also replace the square with something else, for example, an equilateral triangle.