This is not a name that I am connected with but I have been researching Nova Scotia families for quite some time. I was interested in this query because the name Burgess is more common in Hants and Colchester counties where the first settlers by that name were part of a group from the New England states which are commonly referred to here as the New England Planters. (you can google that phrase for more information) and settled here in around 1765 -1770. Settlers directely from Scotland were later and this made me think that your Burgess family probably came to Nova Scotia from New England.
Census records show John A as son of Colin and Sarah Ida Burgess. Census and Vital Stats records show Colin as son of Alfred and Clarrisa and Alfred’s death record, also from Vital stats give his parents as Freeman and Sarah.
I did a search of submitted records on the LDS website and found the following. These of course are not necessarily substantiated but do suggest possible places for researchhttps://familysearch.org/pal:/MM9.2.1/SGXF-JSC
The above website gives the following information
Freeman Burgess born April 17, 1781 in Harwich, Barnstable Mass, son of Jonathan Burgess and Mehitable Chase
Married Sally Kelly Jan 23, 1806 and had:
1. Patrick Burgess born 1810, Ontario
2. Watson Burgess born 1811, Nova Scotia
3. Alfred Burgess born 1812, Nova Scotia
4. Ensign Burgess born 1813, Nova Scotia
5. Josiah Burgess born about 1817, Nova Scotia
6. Marshall Burgess born 1825, Nova Scotia
7. Sarah Burgess born 1823, Nova Scotia
8. Huldah Burgess born 1821, Nova Scotia
9. Freeman Burgess born 1827, Nova Scotia
The 1871 Canadian Census lists Alfred, Patrick, Watson, Freeman and Ensign Burgess as heads of households in the area of Port Mouton which certainly correspond well with the above although a couple of them would be second generation given their age at the time of the Census.
What the above seems to show is that if you are interested in going back further in with the Burgess family Barnstable, Mass could be a good place to start.