Translation:He is not going to have breakfast with me.
The Oxford English Dictionary has citations of "breakfast" as a verb going back to the 1600s. Similarly, there's a song called "Here's to the Ladies Who Lunch" in Stephen Sondheim's great musical "Company." But I wouldn't dream of using these words as verbs in ordinary conversation. As other posters have noted, the words sound formal and archaic.
"to breakfast" would be understood by some, not all, since it's a highly stylized use of English. It's not even colloquial, it's not natural, and you'd be viewed as odd for saying it.
Same to "to lunch".
"to supper" and "to dinner" are never used: "to sup" and "to dine" are the verbs associated with them, but "to dine" is the only one that ever gets used, and then it's a kind of refined English which is not used in common parlance.
They're similar, but not quite the same.
"He won't have breakfast with me" tells us definitely that he won't, while "he doesn't want to" doesn't necessarily, depending on context - he might have to for some reason, but he won't enjoy it.
"He doesn't want to" also tells us something about his motivation - if he doesn't have breakfast with me, it's because he doesn't want to. "He won't", on the other hand, doesn't say anything about motivation. Maybe he would like to, but he just doesn't have time.
"He will not be having breaking breakfast with me"
Duo has an odd habit of using "going to [verb]", a phrasal future borrowed from Spanish, instead of actual future tense "will [verb]" or "will be having breakfast", since "to breakfast" is strange and unnatural English.
I sometimes think that Spanish was the first Duo module created, and the phrasal future (used freely in Spanish) has wandered into the other language courses. It's not incorrect or even bad English - it's just not the same as actual future tense. "going to" and "will" are not exact synonyms.