Cheap oscilloscope showing 16 MHz square wave
I believe there's a problem with the scope. Am I right?


Don't think so. Overshoot is a perfectly normal phenomenon when measuring a fast-edge signal with a high-impedance probe. (Also, these signals look about as sharp as I'd expect them to be.)

There's many tutorials on sensing high-speed signals: this is the perfect time to read one!

Oh, and there's Gibb's phenomenon, which says that any band-limited observation of a theoretical perfect (or far less band-limited) edge will have some 9% of overshoot; to understand that, I'd recommend looking at the cosine series representation of the square wave and consider what you'll cut off when you get rid of anything above 5× 16 MHz (=the fundamental frequency of your square wave).