Both EAT BREAKFAST and HAVE BREAKFAST are correct and are practically interchangeable which basically means that you may use them in the same way and they will mean the same thing.
I usually eat breakfast on my balcony. * I usually have breakfast on my balcony.
To go a step further, verbs EAT or DRINK and HAVE refer to the act of eating or drinking something.
We had a chocolate cake at the party last night. *We ate a chocolate cake at the party last night.
People with high blood pressure shouldn't drink any coffee. * People with high blood pressure shouldn't have any coffee.
If you like to be more precise, it's worth noting that if your breakfast consists of a glass of orange juice or a mare cup of coffee and a cigarette, then you'd opt for HAVE BREAKFAST since you're actually not eating anything.
I was in hurry so I just drank a cup of coffee for breakfast. * I was in hurry so I just had a cup of coffee for breakfast.
NOTE: Of course same rules apply to lunch and dinner or other meals.
NOTE: Of course same rules apply to lunch and dinner or other meals.