This seems to almost always come back to definitions:
Everyman: in D&D terms, it's a term referring to common folk who are NOT adventurers.
Class: in D&D terms, it's a character's profession and adventuring role.
So. Common folk can't have a class. They are what's been referred to in earlier editions as "0-Level NPCs". Like the town guard or a local priest or any of the nobody NPCs who are not and have never been adventurers. Now, if they had been or were adventurers, they would have a class. These are fundamental elements of D&D.
But, a group considering playing a party of common folk could check two sources:
- 5e's Von Richten's Guide to Ravenloft: it has rules for using "Survivors" (low-powered PCs)
- AD&D's Treasure Hunt (N4): it has rules for creating 0-Level PCs