Peter answered how to explain the ideas to students. So I'll answer your second question:
Is "square meter" a badly-phrased term? What would be a more appropriate term?
It's not badly-phrased, but I see what you're getting at: "square meter" is in reference to a particular shape, whereas something like "area meter" would sound more general to other shapes.
However, the problem with a phrase like "area meter" that is intended to be general to other shapes, is that not knowing what specific shape you're talking about actually creates more confusion. Say I ask you to shade in 1 "area meter". What do you do? Do you draw a triangle with sides of length 1 and then shade it in? Or do you do that with a square? A pentagon? A circle with radius 1? The nice thing about "square meter" is that its name tells you exactly how to measure it.
More abstractly, you can view units of area as being a combination of two independent units: "square" tells you the shape, and "meter" tells you the side length. In that view, "square" is the conventional unit for shape, but you could talk about a thing like "1 triangle inch". But using "square" usually makes things way easier for our brains, just like using a decimal number system usually makes things way easier than working in binary.