You might just be a sensitive person who is easily hurt and your body reacts by crying. Even though it's embarrassing, it's better that your emotions are released this way rather than some other way (such as having mental or psychological problems, especially in the future).

When someone hurts us (says mean things, makes fun of us, etc.), I suggest focusing on the person and ask him/her (either out loud or to yourself) what makes that person act that way. It's usually because;
they have (or come from) a dysfunctional home life where they're probably verbally (and possibly physically) abused,
they don't like themselves (poor self-esteem) and
their unhappiness makes them lash out at other people, put other people down, etc.

When someone makes fun of me;
I stay calm (knowing that they're probably in pain, psychologically and emotionally) and
I immediately (so I don't have a chance to cry ) ask them what kind of family life do they have, what kind of childhood did they have (if they're a grown-up).
They usually respond with some more verbal abuse (because now they're scared because I've moved the focus back onto them and this makes them very uncomfortable) and
I respond back to them, using what they said "towards" them. For example, if they say I'm ugly, I ask them do they think of themselves as ugly (because it's human nature that we accuse others of what we're guilty of ourselves, or think we're guilty of ).
I keep talking about them and that usually shuts them up.

omg same, I have these problems as well and for me it started at age 11.

Better not to be sensitive to others' utterances...just hear it and forget it..don't be a cry baby at any circumstances and spoil your own moods.

It's ok to cry. You are not a crybaby. Grownups cry too. It's ok.