I would like to define mindfulness not as a mystical phenomenon but as a scientific approach to achieve thinking directed to one’s own cognitive processes. This topic is well researched by Adrian Wells, who is a clinical psychologist. In simple terms metacognitive therapy attract attention to patients’ internal problems and helps them to resolve these problem. I can also say it that metacognition is the main aspect of critical thinking, because critical thinking is mostly directed to one’s own thinking. So I can conclude that metacognitive therapy is an application of critical thinking.
You can be a critical thinker if you can perceive and have some influence on your mental processes. But you better do that in right time and place, so you should also be mindful to your surroundings and your internal processes. This is related to the problem of skill transfer. A person can use a skill in a classroom, but the same person may fail to use the same skill in another context. Even in case of a similar situation something will be different, so a new context may not have cues needed to remind about necessary skill.
But if a person use mindfulness often and analyzes the environment, he or she can notice a situation when a specific skill should be used. That may sound like very hard task, to be mindful all the time. But as a mental process even mindfulness can partially become an automatic process or at least a heuristic one, when it have been practiced for a while. Like other mental task our brain would optimize mindfulness over time and tries to save its resources making mindfulness easier for itself. So mindfulness can be practiced like other skill and a person can become better in it over time.
Of course mindfulness can be used not only with critical thinking skills but with any other skills. Better if a person has active open-minded thinking mindset he or she can connect different skills and different context together and find novel ways to solve problems.