From this FoD site's PRs:
Could this be him I wonder ??
(Found by use of Advance Search for Baptisms, just inputting forename "Henry John", mother "Emily", no surnames searched. This gave only ten hits for the whole database..)

Record ID 243912
Entry Number 416
Year 1874
Month [not stated]
Day [not stated]
Parents Surname WILLIAMS
Child Forenames Henry John
Fathers Forenames George
Mothers Forenames Emily
Mothers Surname RUCK
Residence Upleadon
Occupation [not stated]
Officiating Minister Joseph White Curate
Event Baptism
Memoranda Son of
Notes Born: Jul 26/74 Privately Baptized: Jul 26/74 Illegitimate No day or month given for received into the church.
Register Reference P346 IN 1/6
Page Number 52
Parish Chapel Upleadon

Unusually I cannot see a corresponding Birth registered abt this time/location, either on FreeBMD or GlosBMD..... given the vague dates, the Notes and the Names I do wonder if the above Baptism was subsequently officially registered as the Birth below ? (I don't know (can anyone help please ?) but as far as I understand the Birth is registered by a parent after the event and could be done without the Registrar knowing if the surnames were "confused" ??)

From GlosBMD:

Birth Details
Child Surname Child Forename Father Surname Mother Surname Mother's Former Name Year District Office Register Entry
MINCHIN Henry John MINCHIN MINCHIN WILLIAMS 1874 Gloucester Gloucester, Newent 17 45

And FreeBMD:

Surname First name(s) District Vol Page
--------------------------------------------------------------------------------
Births Jun 1874 (>99%)
--------------------------------------------------------------------------------
Minchin Henry John Newent 6a 256

Upleadon is but an hours walk from Newent.
http://www.genuki.org.uk/big/eng/GLS/Upleadon/Gaz1868.html
http://www.upleadon-village.co.uk/st-marys-church/

NOW I realise this FoD site carries the 1874 Marriage of George & Emily, but thorough searching of the 1881 & 1891 England & Wales Census for this couple with son Henry John gives no results ?

Record ID 53968
Entry Number 48
Year 1874
Month Sep
Day 24
Grooms Surname WILLIAMS
Grooms Forenames George
Grooms Age 30
Groom Condition Bachelor
Grooms Occupation Labourer
Grooms Residence Upleadon
Grooms Fathers Surname Williams
Grooms Fathers Forenames John
Grooms Fathers Occupation Labourer
Brides Surname RUCK
Brides Forenames Emily
Brides Age 22
Brides Condition Spinster
Brides Occupation [not stated]
Brides Residence Upleadon
Brides Fathers Surname Ruck
Brides Fathers Forenames Robert
Brides Fathers Occupation Labourer
Licence or Banns Banns
Date of Banns
Signature or Mark He marks She signs
Witness 1 Charles Parry
Witness 2 Lucy Painter
Other Witnesses
Officiating Minister Joseph White Curate
Event Marriage
Memoranda
Notes
Register Reference P346 IN 1/5
Page Number 24
Parish Chapel Upleadon

But where are they in subsequent Census ??.
I've searched for Deaths of George Williams abt this time but there are several, a very common name indeed, so no proof there !

===============================================================================

Secondly, I've been searching the various free websites for your Henry John, you have my "sympathies" in this respect as it's clearly a very popular Minchin forename !

I realise it may well not be your Henry but have you seen this record from LDS ?, I thought it a little different than the more obvious Census/BMD stuff, I know the second name's not mentioned but that's not unusual on early Army records:

United Kingdom, Chelsea Pensioners' Service Records, 1760-1913
name: Henry Minchin
event: Pension
event year range: 1760-1913
event place: United Kingdom
birthplace: Gloucestershire
estimated birth year: 1874

???

Of course it may well be the following Henry so not yours, but thought you may still be interested.

England, Births and Christenings, 1538-1975
name: Henry Minchin
gender: Male
baptism/christening date: 22 Jun 1873
baptism/christening place: St. Mary-de-Lode's Church, Gloucester, Gloucester, England
birth date:
birthplace:
death date:
name note:
race:
father's name: William Minchin
father's birthplace:
father's age:
mother's name: Mary Elizabeth
mother's birthplace:
mother's age:
indexing project (batch) number: I02037-3
system origin: England-EASy
source film number: 855618
reference number: 88

England and Wales Census, 1881
name: Henry Minchin
age: 7
gender: Male
birth year: 1874
birthplace: Gloucester, Gloucestershire, England
relationship to head of household: Son
marital condition: Single
profession/occupation: Scholar
address: 159 Westgate St
census place: Gloucester St Mary-De-Grace, Gloucestershire, England
disability:
record type: Household
family history library film: 1341611
the national archives reference: RG11
piece/folio: 2533 / 42
page number: 41
Household Gender Age Birthplace
head William Minchin M 51 Blockley, Worcestershire, England
wife Mary E Minchin F 50 Longborough, Gloucestershire, England
son William Minchin M 16 Gloucester, Gloucestershire, England
daughter Fanny Minchin F 12 Gloucester, Gloucestershire, England
son Etheridge Minchin M 10 Gloucester, Gloucestershire, England
son Henry Minchin M 7 Gloucester, Gloucestershire, England
visitor Julia M Cook F 24 Bok Norton, Oxfordshire, England
assistant Charles H T White M 21 Kidderminster, Worcestershire, England
assistant Henry P Cummins M 21 Manchester, Lancashire, England
servant Louisa Cockyear F 16 Longhope, Gloucestershire, England
servant Jane Boucher F 14 Gloucester, Gloucestershire, England

etc etc

Hope this all helps, good luck contacting Mirrim,
Jeff.

Regards,

Adrienne Minchin.

John Minchin abt 1827 Gloucestershire, England Dymock Gloucestershire

1851
George Minchin abt 1789 Staunton, Gloucestershire, England Head Dymock Gloucestershire
Mary Minchin abt 1789 Ledbury, Herefordshire, England Wife Dymock Gloucestershire
George Minchin abt 1822 Dymock, Gloucestershire, England Son Dymock Gloucestershire
John Minchin abt 1825 Dymock, Gloucestershire, England Son Dymock Gloucestershire
Eliza Minchin abt 1826 Bredwardine, Herefordshire, England Wife Dymock Gloucestershire
Thomas Minchin abt 1819 Dymock, Gloucestershire, England Lodger Dymock Gloucestershire

1861
Johns Minchin abt 1824 Dymock, Gloucestershire, England Head St Peter Herefordshire
Emily Minchin abt 1836 Herefordshire, England Wife St Peter Herefordshire

1871
John Minchin abt 1825 Dymock, Gloucestershire, England Head Newent Gloucestershire
Emily Minchin abt 1835 Bullingham, Herefordshire, England Wife Newent Gloucestershire
Emily Minchinabt 1862 St Peter, Herefordshire, England Daughter Newent Gloucestershire
George Minchin abt 1868 St Peter, Herefordshire, England Son Newent Gloucestershire
Hannah Minchin abt 1865 St Peter, Herefordshire, England Daughter Newent Gloucestershire

1881
John Minchin abt 1825 Dymock, Gloucestershire, England Head Breinton Herefordshire
Emily Minchin abt 1834 Bullingham, Hereford, England Wife Breinton Herefordshire
Emily Minchin J abt 1862 St Peters, Hereford, England Daughter Breinton Herefordshire
George Minchin abt 1868 St Peters, Hereford, England Son Breinton Herefordshire
Henry J. Minchin abt 1874 Oxenhall, Gloucestershire, England Son Breinton Herefordshire
Martha Minchin abt 1878 Oxenhall, Gloucestershire, England Daughter Breinton Herefordshire
Mary A. Minchin abt 1872 Newent, Gloucestershire, England Daughter Breinton Herefordshire

This is not my Emily Minchin so please get in touch if you recognize this ancestor.

Mirrim.