Buck converter: what determines the inductor voltage?
It's all down to Kirchhoff's voltage law. Let's look at a circuit:

Consider the instant the switch is opened. Your input is disconnected, which means:
$$V_\mathrm{out} + V_\mathrm{inductor} + V_\mathrm{diode} = 0$$

This is where the negative voltage is coming from. The output voltage cannot change instantaneously because of the capacitor. This means that the inductor voltage needs to become negative in order to satisfy Kirchhoff's voltage law.