No, not all of us have red hair.
Red hair is a relatively rare hair color, determined by genetics. Specifically, it is most commonly associated with a mutation in the MC1R gene. This gene provides instructions for making a protein called the melanocortin 1 receptor. This receptor helps determine skin and hair color. When someone inherits two copies of the mutated MC1R gene (one from each parent), they are likely to have red hair.
The reference provided discusses some characteristics often associated with red hair due to this genetic link, such as potentially needing higher doses of certain painkillers or anesthesia. This is unrelated to whether everyone has red hair, but highlights some interesting differences that can arise from having the specific genetic makeup for red hair.