Yes. You can show this by SAS of two right triangles. Consider rectangle ABCD. AD and BC are the same length and AC and BD are the same length because opposite sides are congruent. The angles ADC and BCD are congruent since it is a rectangle and the angles are right angles. So the triangles ADC and BCD are congruent and their hypotenuses (the diagonals of the rectangles) are congruent.
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rectangle and parallelogram
A square and a rectangle have diagonals of the same lengths.
In a quadrilateral, the diagonals are only congruent for rectangles (or squares, which is a special kind of rectangle). Note: they are not congruent for a Rhombus.
Regular polygons. A square and rectangle will have congruent (equal length) diagonals.
square, rectangle and an isosceles trapezoid.