Probably not, this is wrong on multiple levels. (Sorry.) :´)
"Bouncer" appears to translate as ajtónálló (lit. "standing in the doorway"). "A kocsma őre" is a possessive construction and it should be translated as that. He's the guard of a pub in particular, and not just "the bouncer".
You can, however, try to say "the pub's guard" if that sounds less awkward to you.
Ajtónálló means specifically someone whose job is to regulate who goes through a door (ie. a bouncer, I guess), but the word itself is very rare. I've mostly heard it medieval contexts (or in a pejorative way sometimes.) (Doorkeeper might be a better translation.)
Security guard is usually biztonsági őr. Their job might involve keeping a watch on the customers inside too.
As a native speaker, the Hungarian sentence sounds sarcastic to me. János spends all his time in the pub, ("in order to make sure no harm comes to it.") Stylistically & meaningwise it's the same as saying that "he technically lives there."
If it is not the owner, nor the bouncer or doorkeeper, - maybe the one who is in charge of the bar? The one (as a paid job), who is doing all kind of work there, like in small villages? Sometimes they are only open for about 2 hours a day. Or is it more one of the drunkards, who is always there?
There's also bár in Hungarian, and it's a bit different from a kocsma. The former is a rather cosmopolitan place, mostly to be found in cities and frequented by people who jumped right out of "Sex and the City".
Kocsma, pub, however, is more casual, cosy, familiar. The kind you find in a village where the old farmers gather when the day is done and have egy sört or five.
You can look at Google's pictures for both terms and get a feel of the difference.
Well, the sentence is supposed to mean that the guard's name is János (who I strongly assume is a he). But looking at it from your point of view, it can also be interpreted as if the speaker is talking to János about the guard, who's a different person. Your translation is valid. :)
If I talk to someone about Janos, the guard, wouldn't I just say: "Janos a kocsma őre", if I mean it sarcastically "A kocsmaé János, ő a kocsma (az ő ??) őre" (Janos belongs to the pub, he is its guard) and if I talk to Janos about someone else as guard I'd expect the given sentence.
You cannot use these sentences if your dialogue partner doesn't know who János is. The original sentence is basically an introduction, following the common pattern of "This is the [job description], [name]." This is the janitor, Stephen. You can also do it the other way around, saying "Ő János, a kocsma őre."
Your second sentence could translate as "János a kocsmáé, (ő) az őre."