Prior to 1864 and civil registration of births in Ireland, unless the precise family history has been documented and passed down through the generations, you will have to rely on church registers to confirm birth details, including names of parents, of Andrew Ritchie and Mary Jane Boden born c. 1802 possibly in the wider Londonderry/Derry area. Dates of commencement and quality of information in church registers vary from parish to parish and from denomination to denomination. Access to church registers, in the absence of indexes and databases, is generally gained through knowledge of an ancestor's parish address and religious denomination.
'Londonderry' or ‘Derry’ as a place of origin can refer to the city, the county or port of departure. In the 18th and 19th centuries the catchment area of port of Londonderry for emigrants departing Ireland were Counties Londonderry, Donegal and Tyrone and to a lesser extent northwest Antrim and north Fermanagh. In theory, your ancestor could have originated in this wider area.
There are 45 parishes in County Derry. You can identify the civil parishes of County Derry, and their associated townlands at https://www.johngrenham.com/places/civil_index.php by
selecting Derry on the map.
There is no national index to Irish church registers. To date, only the county-based genealogy centres have attempted any large scale, systematic indexing of church registers in their localities. Although RootsIreland, https://www.rootsireland.ie,
is the largest online source of Irish church register transcripts, it must be emphasised that a failure to find relevant birth/marriage entries in this database (or indeed any of ‘big’ Irish databases) doesn't mean that the events you are looking for didn't happen in Ireland. It simply means that they are not recorded in the database; for example, they may be recorded in a record source which doesn't survive for the time period of interest or in a source that has not been computerised.
Although Derry Genealogy has computerised the baptismal, marriage and burial registers of 85 churches in County Derry (26 Roman Catholic, 24 Church of Ireland and 35 Presbyterian) at http://www.derry.rootsireland.ie
(the earliest being St Columb’s Cathedral in Derry city, dating from 1642), only 17 of them have any surviving registers prior to 1810. Hence the birth, marriage and death details of many of our 18th and early 19th century ancestors have not survived in a written record source. It is quite possible that the births you seek occurred before their written confirmation in a surviving church register. I can confirm that this database doesn’t record birth entries for either Andrew Ritchie or Mary Jane Boden. The church registers computerised by Derry Genealogy are listed at https://rootsireland.ie/derry/online-sources.php.
If you wish to identify all available church registers in the wider Londonderry area, microfilm copies of which are held in our national archives in the Public Record Office of Northern Ireland (PRONI) in Belfast you should explore their Guide To Church Records, which can be accessed on their website at https://www.nidirect.gov.uk/publications/proni-guide-church-records
as it lists, in alphabetical order by civil parish, church registers of all denominations for most parishes in Northern Ireland and their commencement dates, together with their microfilm reference details. PRONI (contact detailsat https://www.nidirect.gov.uk/contacts/contacts-az/public-record-office-northern-ireland)
is the major repository of public and private archives for the six counties of Northern Ireland. You can examine any records held in PRONI, at no charge, if you visit this office. If you require a researcher to examine record sources on your behalf you would have to pay for their services.
In searching for Presbyterian ancestors, you should also check the registers of the Church of Ireland. It is possible for the baptism, marriage and burial records of Presbyterians to be recorded in the registers of the Church of Ireland, as for centuries the Established Church (i.e. Church of Ireland) claimed the right to administer baptism, marriage and burial ceremonies to all Protestants.
The port of Londonderry was the major emigration port for northwest Ireland to North America (i.e. counties Derry, Donegal and Tyrone) from c. 1680 to 1939. To the best of my knowledge there are no surviving passenger departure lists for Irish ports or arrival records for Canada c. 1827. Local newspapers of the time such as the Derry Journal, published since 1772, would have recorded ship departures and carried advertisements for ships seeking passengers for North America, but didn’t, as rule, record passenger names.
Saint John (New Brunswick) and Quebec in Canada, and New York and Philadelphia in the United States were the destination ports of emigrants departing from Derry in the first half of the 19th century. For example, of 38 emigrant ships advertised in Derry Journal to sail from Derry in 1836: 12 were destined for Saint John (New Brunswick), 12 for Philadelphia, 7 for Quebec and 6 for New York.
Although Protestant migration as a proportion of Irish migration declined relative to Catholic migration after 1815, the numbers of Irish Protestants emigrating to North America in first half of 19th century was still greater than numbers that departed in 18th century. I think it is fair to say that throughout 18th and 19th centuries many small farmers, agricultural labourers and rural tradesmen in Ireland, whether Catholic or Protestant, saw emigration as the only solution to their declining economic prospects.
Quite often the only realistic strategy in tracing ancestors beyond church registers (which are the building blocks of family history) is to examine surviving land records and census substitutes, often compiled by civil parish, for any references to a surname or given name of interest.
County Derry, for example, is fortunate in that an abstract of the 1831 census has survived. This census, arranged by parish and townland, lists: the name of each head of household; the number of individual families in each house; the number of males and females, including servants, in each household; and their religious denomination. Although this source lists the names of all heads of households, 40,000 in total, in County Derry it doesn’t name the other family members within each household.
This source which can be examined at https://billmacafee.com/1831census.htm
names 7 Boden households and 51 Ritchie households residing in County Derry in 1831. Although your ancestors had emigrated by 1831 it is quite possible, but not proven, that they are connected with some of these households. However, it must be emphasised that as this source only names heads of household it provides insufficient information to confirm the nature of the link between any of the Boden and Ritchie households recorded in 1831 census returns and your ancestors.
With best wishes