I strive for one gram of protein per pound of body weight. I also drink lots of water and eat lots of vegetable and fruit.
To answer your question, I rely on whole foods as much as possible. But if I need to up my protein intake, yes I will drink a shake on rest days. Remember, rest days are when your body heals and becomes stronger from your training. So nutrition is a 24/7 proposition.
I find that I have much more delayed onset muscle soreness if I don't have my protein shakes, starting from the hour after the workout until as many as 3 days later. Recovery is where you grow and get stronger, so make sure to fuel yourself while that is happening.
Whey protein probably wouldn't be the best option for an all around rest day protein. The reason why people use whey post workout is because of its fast absorbtion rate. For a recovery day, I think "real" proteins would be more beneficial. Proteins from dairy/meat/legumes will take longer to digest in your body, which means the absorbtion will be a bit more spread out and you can utliize more of the protein. Take your vitamins too.