A generalization of the "Four vertex theorem"

You can prescribe Gaussian curvature function on the smooth sphere as long as it is positive (Kazdan and Warner, 1980s) and then embed the surface isometrically as a convex surface in $E^3$: Alexandrov, Pogorelov, Nierenberg. The history of the embedding result is a bit convoluted: It was first proven by Alexandrov in 1940s without any smoothness properties; smoothness (as well as uniqueness of convex embedding) was established shortly thereafter by Pogorelov and few years later by Nierenberg. Nierenberg's proof is probably the most readable. I will chase the references when I have time.

Thus, the answer to your question is negative, you can have exactly two critical points. It would be interesting though to find out if your question about critical points has a positive answer for the mean curvature function.