Is it possible to draw 3 straight lines through a square such that they divide it into 7 equal-sized regions?
No, it's impossible.
Buck and Buck, "Equipartition of Convex Sets", Mathematics Magazine 22(4) (1949), pp. 195-198, showed the following:
"We will be concerned here with three cuts; in general they divide a convex region into seven parts. If the cuts are concurrent, six parts result. We prove that division into six equal parts is always possible, but that although some regions can be divided into seven equal parts, no convex region can be so divided."
For the square,
since it is convex, it **cannot* be divided into seven equal parts by three straight lines.