"Have a breakfast" or "eat a breakfast" in AmE

Usually, we omit the a in a situation like this:

I'm going to eat/have breakfast.

Unless you want to say something like this:

I usually eat a [adjective] breakfast.

Or:

We're having a breakfast meeting.

Etc.

But in the specific case you asked about, eat and have are interchangeable. They both convey the same message.

According to this Ngram, have is more common, but just barely.

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.