Using m-dashes for garden variety appositives is overkill. They're perfectly happy being separated by commas, but only when necessary.
If you have more than one foster brother, then John's name is an essential part of the sentence and has no commas:
My foster brother John is taking a trip.
If John is your only foster brother and his name is only additional information, then it's commas all the way:
My foster brother, John, is taking a trip.
John, my foster brother, is taking a trip.
m-dashes are used for parenthetical information or an aside that breaks the flow of the sentence far more than someone's name:
My foster brother John was unusually tired today — he hadn't slept in at least 36 hours — but he managed to stay awake during the entire film.
Since you've chosen the longer construction place of his birth rather than birthplace, your sentence is perfectly made for another punctuation mark: the colon.
John, my foster brother, and I made a sentimental journey to the place of his birth: Armenia.
This is effective only if you plan to spend a sentence or two talking about Armenia. If you're immediately going to talk, say, about the people you met there, then use:
My foster brother John and I made a sentimental journey to his birthplace, Armenia.