Our body isn't made of clay. Their chemical compositions differ way too much.
The composition of clay and human body is different. Clay consist to a large part of silicon (Si), about 40% normally. A human body doesn't have that much silicon at all, but has instead large amounts of oxygen (which, granted is usually bound to Si in a SiO2 molecule, but not enough to account for the amount in the human body), but also carbon, which is minor, or even non-existent, in pottery clay.
Ergo, you can tell from that sentence in the Quran compare it to our scientific knowledge, and understand that it's not to be taken scientific or literal since clay and human body composition is fundamentally very, very different.
Basically, the only way clay can become human, is to harvest a lot of clay, remove some components radically, or almost completely, and extract others to concentrate them, in the end, you don't end up with clay or pottery clay at all. You can do this the same way with dirt, dust, polluted air, sea water, if you just harvest enough (tons and tons) to extract some parts and discard others, which doesn't make a case for being made out of "clay" any special, but rather it shows that the idea that we come from earth, nature, is the basic concept here.
That verse really means to say, "God made us from nature." Just as evolution says. Evolution only explains how we were made from nature. Simple as that.