Have breakfast is correct.
It is one of the article use rules (or rather a set of exceptions) not to use the indefinite article with meals: breakfast, lunch, dinner, supper, brunch.
Keep in mind that if you are adding an adjective to describe it, then you need to add one: I had a quick lunch, we are having an early breakfast, we had a delicious dinner.
Of course, if you are referring to a specific meal, you need to use the definite article: I couldn't make it to the dinner, the breakfast they served wasn't great, the lunch went well.