The problem.
Somebody has said something wrong, or mean, or absurd, yet they are a parent or a teacher or a boss. I won't say anything to them, and it eats at me. I can't tell them how I really feel. It makes me feel fake. I smile when I really don't want to.
The Zen understanding.
For a Zen Buddhist, a primal reaction or feeling is not considered how we "really" feel. If someone were to lash out at a Zen Buddhist, and hit them, and scream, "Tell me what you really think about me!" the Zen Buddhist would say with the utmost honesty, "I really think you are a buddha."
Sure, we might get angry internally, but if we react angrily, that is a lapse. Our higher reasoning is what makes us human, not our animalistic urges. Because it is the lowest common denominator in our species, many people think that being bold and angry is being genuine. But for a Zen Buddhist, who strives upward, our peacefulness is genuine. This genuineness is founded in reason, in introspection, in the analysis of our practice. For a Zen Buddhist, reality is love.
At some level, most humans realize that they want love and peace. The clearer-minded a person becomes, the more peaceful they become. On the other hand, the more deluded a person becomes, the more violent they become. "Peace and love" isn't something silly people say. For a Zen Buddhist, peace and love is a logical and highly immersive religion. You will not find sharper or more concentrated students.
So for a Zen Buddhist, when we become upset, smiling and being patient is how we really feel. We always keep our interlocutor in mind, and do what think will promote peace. Sometimes, though, this does mean standing up to someone.
Thus, a Zen Buddhist is like a rounded object: when we have reasons to believe, we are firm and even, but when confronted we are also edgeless. A Zen Buddhist strikes balance. We do not foster resentment with the use of harsh words, nor indulge the ego with spinelessness. We foster altruism. Through patience and quiet teaching, a Zen Buddhist helps others find what we have: happiness. This is the bodhisattva's way.
The Zen application.
When a Zen Buddhist gets upset, it is time to calm down. Never speak hastily, even if that means taking some time alone. Breathe deeply and focus on that breathing. A Zen Buddhist knows that, no matter how horrible a person is behaving, that person is a part of us, a part of all of us. A Zen Buddhist does not want anyone to suffer.
Anger is powerful. When someone becomes angry, people shy away from them. The world around that person becomes terrible; they become like a storm cloud. Therefore a Zen Buddhist avoids angering. To accomplish this, we study our anger, accept and care for it, and then allow it to fall asleep like a child. Our violent emotions are animal-like, childlike. They are not a reflection of how we really feel, no more than a child is wiser than an elder.
A Zen Buddhist is not a wrestler, a slayer of negative emotions. A Zen Buddhist is a mother, someone who understands negative emotions and looks after them while they slumber.