The size of the pan matters
If your pan has the same width as the diameter of the round pan in the recipe, it will be too big. If you imagine a circle drawn inside the square pan, you'll see that a large amount of batter will be in the four corners. If you're good at geometry you can figure out the area of a circle with 18 cm diameter you can figure out how very small that square cake pan would have to be to make the cake rise properly.
The method is not described in your recipe, and the method is everything. Egg yolks beaten with most of the sugar, egg whites beaten with the rest of the sugar, flour and butter carefully folded in with least movement, these are essential. But in the wrong size pan, you are spreading your batter out so it can only be like a thick pancake.
Bughut, i don;t think if it's made properly it would need baking powder, just look at the size of the pan, it's very small. 7 inch, not the usual 8 or 9 inch. Beaten egg yolks, beaten egg whites, proper folding,
and ChicagoTerry, i think maybe the 2 eggs (large eggs, perhaps) would be enough for the same reason. I would use 3 eggs for an 8 inch round pan, so 2 might be enough. But yes, the technique is very important. But mainly the size of the pan is essential. I doubt they even make a square pan small enough to be equvalent to a 7 inch round one.