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I am doing an assignment on vectors and geometry. And I would like to know if such a property exists for squares.

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I have been watching that your question being left unanswered for quite some time. Though I tried to understand it, wasn't sure how perfect am I. So could you kindly rephrase your question please?
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The answer depends on which two mid points are you choosing on the square.

If you are choosing mid points of opposite sides then its length will be equal to the length of side of the square.

But, suppose you choose the mid points of two adjacent sides on square. Let the length of side of a square be of a unit. Then the length from one midpoint to another on a square will be a/(sqrt2) while half the length of side is a/2. Therefore these two are not equal.
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