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1st battalion 14th Regiment of Foot aka the Prince of Wales's Own and the Buckinghamshire Regiment.
Embarked at Queenstown 19th August, arriving Malta 29th August 1867. Cholera outbreak on arrival. Remained in Malta until late 1868 when it went to India, according to information on these websites.
14th Regiment of Foot (The Prince of Wales's Own)
https://wiki.fibis.org/w/14th_Regiment_of_FootChronology + time in India.
"Regiments of the Malta Garrison
The 14th (Buckinghamshires)"
www.maltaramc.com/regmltgar/14th.html Try "Stations of the British Army" for each year 1862-67 to track the movements of the regiment. List of regiments and their locations were published in some newspapers e.g. Edinburgh Gazette. I don't know how often the lists appeared. I looked for a few random years.
e.g. 1863 A March edition of a Welsh newspaper gave the information that 14th Foot 1st battalion was in Jamaica and that 2nd batt. was in New Zealand.
https://newspapers.library.wales/view/3093490/3093497 Searching for Stations of the Army for other years shows 1st batt. was in Jamaica on a list published Dec. 1862, at Aldershot August 1864 and Sheffield at some point in 1866. 2nd batt. was in New Zealand for a much of the time. List in "Edinburgh Gazette" dated 1st Jan. 1864 has 14th Foot in Halifax N.S. (Nova Scotia?) but doesn't specify which battalion.
It seems likely that John spent most of the period 1862-1867 outside Ireland.
There are several websites about the army in Ireland.
"Garrison Towns" is worth exploring.
https://irishgarrisontowns.com There are websites devoted to the army in County Cork. An internet search for terms such as "British military barracks in County Cork" and "British Army County Cork" + a range of years should find some.
Finding information about early lives of John and Bridget will be harder. Birth registrations didn't begin in Ireland until 1864. Before then the main source for births is baptism registers. To find people in those you need to know where they were born, roughly when and their religious denomination. You also need Bridget's maiden surname. Preferably also name/s of parent/s of John and Bridget. Registers in many R.C. parishes didn't start until later in the century. Some C. of E. registers were destroyed or badly damaged in 1922.
Have you found them on censuses? What information did they give about place of birth? Were they consistent about age?
Were there other children after John? What were their names, approximate birth years and places of birth?
Was their son John baptised and if so, where, when and which denomination?
A guide to Irish family research is Irish Genealogy Toolkit.
https://www.irish-genealogy-toolkit.com