Question

What quadrilaterals must have all four sides of equal length

Answer 1

Squares and rhombuses/diamonds.

Answer 2

An equivalent condition is that opposite sides are parallel (a square is a parallelogram), that the diagonals perpendicularly bisect each other, and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (four equal sides and four equal angles).