I've given it some thought and maybe have 'half a solution'. :)
I calculated the area of an equilateral triangle of unit length (=1) as √3/4 square units. [The height is √3/2 (Pythagoras), so the area is √3/4.]
The area of the square must be the same, so the length of the 2 sides of the square must be 4√3/2 [4th root].
Half the height of the triangle is √3/4, So the square root of half the height, will give us 4√3/2, which is the length of one side of the square. Once we've got that then we can cut everything to fit.
I then went down a bit of a rabbit warren trying to work out how to construct that, perhaps using a compass. I'm sure it's possible but don't have the urge to spend any more time on it.
How close did I get?