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The sum of the squares of the sides of the polygon is the same as the sum of squares of the sides in right angled triangles corresponding to each side of the polygon, as constructed in the picture below.

It is clear from the picture that the sum of those (blue) sides are at most 4, and that none of those blue sides is greater than 1.

From that follows that 4 is the maximum sum of their squares.

This is illustrated in the picture below, where a rectangle of size 1 × 4 is covered by squares of which none is greater than 1. Clearly it is impossible to cover more area than the rectangle itself.

Thus the sum of squares of the sides of the polygon is also not greater than 4.