As you have observed, the eating of the first meal of the day is nowadays mainly expressed in terms of consuming breakfast as a noun: to have breakfast, eat breakfast, or get breakfast, or more formally take breakfast, or more colorfully devour, scarf, etc.
You can use breakfast as a verb. The dictionaries still cite this sense, e.g. Oxford Living Dictionaries:
verb [no object]
‘she breakfasted on fried bread and bacon’
I would caution, however, that this usage is somewhat dated for conversational English. A search on breakfasted, for example, against various BYU Corpora turns up mainly examples from literature or long-form journalism, albeit some in columns written in the first person. While this is no proof of non-usage (those are the media the corpora mainly index, plus it's hard to prove a negative), I would certainly say that among Americans, using breakfast as a verb in conversation sounds pompous. It is the kind of affectation you'd read in a resort brochure, or expect from a pre-war butler in a film. I would say the same is true of lunch and sup as verbs as well— though nothing is worse for that than seasons (e.g. We summer in Banff and winter in Tamarindo).
Dine and dining, in contrast, have taken on additional meanings and connotations, and are not necessarily tied to whichever meal is called dinner in a particular region. There is little pearl-clutching over taking brunch in the dining room, or referring to a high-end, full-service restaurant whose signature dishes are lunch foods as fine dining.
To break fast is now primarily used to refer to the breaking of a religious fast. Indeed, the Wikipedia article at break fast covers Jewish practice (the Muslim counterpart at Iftar, Arabic for "break fast"). You can find references to breaking fast referring to the daily meal of breakfast—
It was Europe's introduction to chocolate, Anderson argues, that helped to change people's perspective on the moral propriety of breaking fast in the morning hours. (Garber, Megan. "The Most Contentious Meal of the Day", The Atlantic, June 6, 2016)
— but they are again exclusively literary, not quoted conversation. While an educated person would understand the meaning if you invited him or her to break fast with you, s/he would probably find it unusual, and if you told the average customer in a breakfast restaurant you were eager to break fast, they'd probably think you were making a bad joke.