square
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Yes! It's called a rhombus.
rhombus
A square has four sides so it is a quadrilateral. All sides of a square are of equal length so it is equilateral. All angles of a square are of equal measure so it is equiangular. A quadrilateral that is equilateral and equiangular is a regular quadrilateral.
A square would fit the given description
No, an equilateral triangle has to be equiangular, but an equiangular triangle does NOT have to be equilateral