*"That's sort of my point, though?
Teaching 'this is a square' and 'this is a rectangle' and 'this is not a rectangle, it is a square' is teaching them that a square is not a rectangle.
Then later you teach them how to define square and rectangle, and they learn that a square fits the definition of rectangle.
So why do it in that order?
They learn something incorrect, then they have to re-learn it."*
You can't not teach them the word square until they're ready to learn all the definitions and properties. It's a word in general use, and many children will have met it before they even get to school.
I'm a maths teacher, and I'm quite happy that young children learn to sort and name shapes without realising that some of them fit more than one definition. They will then be quite happy with those terms when we get to defining them later on. I would like to hope that all primary teachers know that a square is a special sort of rectangle, so that if they get the odd child (probably mine*) who says "but isn't this a rectangle too?" they can have that conversation with them.
By the way, someone mentioned that triangles can be equilateral, isosceles, scalene or right-angled. That's a dangerous one, too, as right-angled triangles are always also either scalene or isosceles. Too many people think there are four sorts of triangle!
*Mine definitely had a dispute with the pre-school over the shape she called a rhombus! But again, diamond is the everyday term, so one to be learned, especially if you're ever going to play cards - it would just have helped if the adults had recognised the technical term!