Can I sample at Nyquist rate if I know that my samples are taken only at the signal's maxima or minima?
when you have a quantized signal (like in your case) you also have a noise floor in the frequency spectrum. This noise floor is proportional also to the sampling rate. If you sample a signal at twice its maximum frequency you can distinguish all the frequency components above the noise floor. In your case you are sampling at $F_p = 2$ because there are 2 samples every second therefore you are assuming your signal has a maximum period of 1.
There is one case when you are interested on sampling at the same frequency of the signal. For example if you know that your signal has an unwanted ripple at the frequency $f_{ripple}$ you can sample at this frequency $F_p = f_{ripple}$ preserving the lower frequencies while the ripple is transformed into a constant error which basically depends on the ripple waveform.