Are rest days important?

This answer depends, but in your case, unless your walks are highly strenuous (steep hills, fast walking, rough terrain), then I don't think a purposeful rest day is "needed" (unless your body is really telling you otherwise).

Now, if you are doing hard walks/hikes, enough to make yourself sore afterwards, an easier day may not be a terrible idea.

Harder workouts, weight lifting, etc - those are more likely to require some rest periods. For example, I lift heavy 5 days a week (plus cardio), and my weekends are definitely needed "rest" days. One of those is generally an active day (house cleaning, horseback riding, maybe a bicycle ride), the other tends to be a bit lazier but may still do some of the above - or not. Right now, I need those days, and by Monday I'm ready to hit the gym again, but come Friday, I'm pretty beat and need the rest.