It helps to understand that in the U.S., "John Doe" is a generic name used to refer to an unknown or unidentified person. You'll hear it used regularly in crime or police shows to refer to unidentified dead bodies.
There is a grammatical difference between your two examples.
Everybody is gonna cut himself a nice fat slash off John Does, eh?
would normally refer to an individual, the "s" (John Does) reflecting a dialect (old Chicago) or colloquialism.
Everybody is gonna cut himself a nice fat slash off the John Does, eh?
would normally refer to a group of unidentified people.
However, in the context of Meet John Doe (1941), the idea of "John Doe" is used to represent the homeless or unemployed, which develops into a political "movement." Thus, there are many "John Does." From this point of view, the two examples you've provided have an identical meaning, the first example being grammatically more colloquial (less formal or specific, perhaps reflecting a dialect) than the second.