Nah, nah   ... I don't think the "James Barrowman" one is yours.   

     [I'm sure your one is just "plain" James HENDERSON - no middle name).  
Had just posted the info re: Thomas Herbert / Thompson Herbert" to show that they were two different people and that Thompson Herbert HENDERSON although buried in the area where your James lived, was less likely to be a son of "a John and Jane HENDERSON": 
But don't splash out on buying any printouts (certs.) just yet.    

I'll check the death microfiche also (for place of registration) - have narrowed deaths down to a handful of likelies.
Found this on paperspast
Otago Witness , Issue 458, 8 September 1860,  Page 4    Article view 
Shipping News. 
Early in June [], the Henrietta and Robert Henderson, [was] despatched from the Clyde by the agents of the Provincial Government of Otago, with full complements of passengers.   Besides those assisted by the agents, a large number not only paid their own passages, but took out with them considerable sums of money.
ARRIVALS. 
3 Sept 1860    Arrived in Dunedin from Glasgow aboard the Henrietta & Robert Henderson 
..............John Barrowman and wife........... William Henderson, Agnes Henderson,..............John Henderson and wife 
DEATHS [on Board]........ June 26, Ann Henderson aged 20 years, of scarlatina...........
this then could be maybe a cousin or brothers that came out together,do you think?
karen