Solution: (1) As shown in the figure, with B as the origin and BC as the X axis, a plane rectangular coordinate system is established.

Then m (2 2,4), h (4 4,0).

Let the analytical expression of parabola be y = AX2+Bx.

Then 4a+2b = 4 16a+4b = 0,

∴a=？ 1b=4。

∴y=-x2+4x.

(2) Let the side length of a square be m, and the coordinates of point f be (6m, 5m).

Point f is on a parabola,

∴-(6-m)2+4(6-m)=5-m.

Finishing, m2-9m+ 17 = 0.

∴ m 1 = 9+ 132 > 6 (s)，m2=9？ 132.

∴: The side length of a square is 9? 132.

(3) Point M cannot fall on BC.

∫ The distance from point M to BC is 4, and the distance from point F to BC is 5-9? 132= 13+ 12,

And 13+ 12≠2,

∴ Fold the rectangular piece of paper in half along the straight line where FG is located, and the point M cannot fall on BC.