John Campbell

Statutory birth, death and marriage registration (in some jurisdictions called Vital Records) only started in Ireland in 1864, save for non RC marriages which were recorded from 1845 onwards. So you probably won’t find statutory birth, death or marriage certificates in Ireland for this family. For earlier years you usually need to rely on church records, where they exist. You obviously need to know the precise denomination in order to search the correct records. Not all churches have records for that period and not all are on-line. 

RC records are mostly on-line on the nli site:   

https://www.nli.ie/en/family-history-introduction.aspx

For other denominations, the churches usually hold the originals but there are also copies in PRONI, the public record office, in Belfast. A personal visit is required to access them. Access to the records there is free. This link explains what records exist, parish by parish:

https://www.nidirect.gov.uk/publications/proni-guide-church-records

If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net

The name Campbell is very common in Co Down as are Hugh & John, so care will be needed to find the right family. (There were 52 people named Hugh Campbell and 142 named John in Co Down in the 1901 census). Here are some Campbells in Co Down to look at. They might include your family:

http://freepages.rootsweb.com/~rosdavies/genealogy/SURNAMES/C/CampbellAi.htm

Farmers would normally be listed in the tithe applotment records. The records for Bangor (1834) don’t list any Campbells:

http://www.irishgenealogyhub.com/down/tithe-applotment-books/bangor-parish.php

Possibly your family came from a parish near Bangor, rather than Bangor itself. For example there was one in Ballyhaskin townland, which is in the parish of Donaghadee, not too far from Bangor.

http://www.irishgenealogyhub.com/down/tithe-applotment-books/donaghadee-parish.php

3 Campbell farms there in 1901, all Presbyterian (so they’ll have originated in Scotland):

http://www.census.nationalarchives.ie/pages/1901/Down/Carrowdore/Ballyhaskin/