Which expression do Americans prefer, have a breakfast or eat a breakfast?
Usually, we omit the a in a situation like this:
Unless you want to say something like this:
But in the specific case you asked about, eat and have are interchangeable. They both convey the same message.
Note: I'm not actually American, but seeing as how there isn't much variation between American and Canadian English, I would say I can still answer this question. I also used the American English corpus for my Ngram.