Vosges is the department, but I can't work out where Château Carré would be. It may refer to a certain house/estate. You need to work out which commune this would have been in as that will be where the birth registration is, if he was entered into the French records.
What was his occupation and the occupation of his father?
Did he marry Florence Augusta M. Boyrenson from India? If so, in 1871 she is living with what appears to be John's sister, Gertrude Fitzmaurice, 29, b. Swinton Yorkshire. Meanwhile, John is a visitor in Nottingham, 33, b. Nancy, France (visiting the Swan family).
In 1861 there is this family:
RG9/777 70/136
Drayton Green Lane, Ealing, Middlesex
Jno Fitzmaurice, 66, Col Half Pay, b. Ireland
Frances Maria, 57,b. Barnbro Yorks
Emma, 34, daughter b. Doncaster
Geraldine, 29, daughter b. Malta
Gertrude, 19, daughter b. Swinton
Agnes M Clyde, niece, 35, widowed, b. Ireland
Margaret F M Clyde, grandniece, 15, b. Van Diemens Land
John Gray Clyd ,grandnephew, 13, b. Van Diemens Land
(+ servants)
(By 1871 Frances, Emma, and "Gertrude" b. Malta == Geraldine? are living together at Drayton Green and it seems John has passed away)
1841 census:
HO 107/1325/12 folio 26 page 23
Silkston, Yorkshire
Henry Watkins, 65, clerk
Frances, 60
Emma, 20
Frances Fitz Maurice, 35
Maria, 15
Emma, 14
Elizabeth, 9 (birth "Foreign")
Henry, 7
John, 3 (birth "Foreign").
As there seems to be a possible army link, I would first look at the GRO's overseas records (regimental indexes and army chaplain indexes) to see if there was an army birth registered for him. Also, the family seem to have been well enough off that there may be newspaper records of them.
There may be a book about this man (the father) - Biographical sketch of Major General John FitzMaurice, K.H, written by Gerald Fitzmaurice in 1908. Also he may be the one listed in A History of the Peninsular War (I can only get tiny snippets on Google Books) It would be worth tracking copies of these books down.
I did look on the Departmental Archives of
http://www.archives.cg54.fr/ for Meurthe-et-Moselle (where Nancy is), but it seems their online services requires payment. As there is some confusion about where exactly John was born (and I cannot find him in 1861 and do not have access to the 1851, so can't help there unfortunately), I would look for UK sources to confirm the exact birthplace before trying to tackle French records-- especially as I'm not sure that his family would have bothered to register him with the French authorities if they were only temporarily residing in the area.