John Flood & Sons definitely did exist, and still does exist as John Flood & Sons (1961) Ltd. That is, it was re-incorporated in 1961. That it existed much earlier is indicated at:

http://www.saintjohn.nbcc.nb.ca/~Heritage/NorthEnd/Homes.htm

which deals with prominent homes and has a photo of employees of JF&S in 1922 working on Holy Trinity School. The names, Burpee and Estabrooks, are well-known New Brunswick names, especially along the Saint John River Valley. I didn't know the Estabrooks were involved with Red Rose Tea though. I thought some of them had money.
The president and CEO is J. Timothy Flood. He seems to be very prominent in construction circles. Some addresses, etc. are:

Mr. T. (Tim) Flood, President
John Flood & Sons (1961) Limited
P.O. Box 2210, Station C
Saint John, NB E2L 3V1
Tel: (506) 634-1112
Fax: (506) 634-0952
tflood@floodconstruction.com

I would doubt that his personal memory would include events in 1938 (?) and since the company has been recreated, there may not be records for that period available.
I saw a reference to post-World War II housing in Fundy Heights in Saint John which stated that John Flood built 106 houses there at a cost of a half million dollars. About $5,000 per house. I have seen similar houses in the Devon area of Fredericton. They were small, based on 5 or 6 basic floor plans, and probably cost a great deal more now. The company is, and presumably was, involved in bricklaying. I would guess this would mean the manufacture of brick and tile, which would involve a kiln saw.
It is possible that the original John Flood was b. 1888 in the City of Saint John, the s/o Matthew and Rebecca (Beamith) Flood. There was, as early as 1893, another John F. Flood who m. Elizabeth Mary Nash, and they had a son, John Nash Flood (who m. 1920 Stella M. Clayton), in Saint John at that time. Still another John Flood m. 1892 Isabella ('Bella') Moore, and they had a child in 1893. A John J. Flood m. 1921 Violet A. Meisner, and this may have been the John James Flood who m. 1915 Elizabeth E. Jamieson (ca. 1893-1919.) A John Michael Flood m. 1909 Catherine Ellen Kennedy.
The John Flood (b. 1859, d. 1902) who m. Isabella Moore was a barber in 1901, so not involved in construction at the time.
Matthew Flood (b. 1832, d. 1907), father of John Flood b. 1889, was from PEI, and a labourer in 1901. Somewhat inconclusive. Not all of the 1901 census for Saint John Co. has been done yet.
Perhaps more likely was was John Flood (b. NB ca. 1852), a mason, with a wife, Mary, in Queens Ward of Saint John in 1881. These give the notion that there were several Flood families, with several John Flood's, although the name was not particularly numerous in Saint John. In 1865-6 a Michael Flood (b. Ireland ca. 1817-21, d. 1902) lived in Saint John and was a mason. He was there in 1871 as well; and may have been the father of John Flood b. ca. 1852. There is a suggestion of a family tradition of involvement in the building trades.
Not a lot, but it shows that the company you father remembered existed at least from the early 1900's, probably had its origins in a family of Irish masons in the mid 1800's, and still exists prominently today in a revised form. It gives you something to focus on. If, for example, you were able to access microfilms of Saint John newspapers of the period, there might be an index.

I don't know if I mentioned it before, but James Isaac Hunt was given as James Edward Hunt in his death record.

Index to Death Certificates: 1920-1951
Name: HUNT, JAMES EDWARD
Sex: M
Date: 1938-05-21 (YYYY-MM-DD)
County: ST JOHN
Registration: 16948
Volume: 116
Microfilm: F19345

I wonder, then, if he was the same in the marriage:

Index to New Brunswick Marriages: 1847-1951
Name: HUARD, MARY ROSE
Married: HUNT, JAMES EDWARD
Date: 1927-04-23 (YYYY-MM-DD)
County: ST JOHN
Parish: SAINT JOHN
Number: 3212
Reference: B4/1927
Microfilm: F19692

Since he and Annie Christenson had a son, James Edward Hunt, it is possible that Edward was another middle name; perhaps James Isaac Edward Hunt. Or perhaps it was the son, James Edward Hunt, who married Mary Rose Huard and was killed in 1938?
According to a web site at:
http://www.theshipslist.com/ships/passengerlists/jane1834.ht...
an Edward Hunt (labourer, age 40), his wife, Biddy (age 31) and son, Edward (age 2) arrived in Saint John on the ship, 'Jane', JUL1834 having sailed from Galway, Ireland. A similar site at:
http://www.theshipslist.com/ships/passengerlists/sjlists1833...
shows the ship, 'Legatus', arriving from London, England with a widow, Margaret Hunt (40) and three children: Jno. Hunt (16), S. Hunt (15) and E. Hunt (13).

I get the sense that:
JAMES ISAAC HUNT b. 11MAY1883, in Dalhousie, Restigouche Co., NB in 1901, m. 1st? ANNA ('ANNIE') CHRISTENSON (CHRISTENSEN) (b. 12JUL1885, sister of Chris Christensen, in Drummond Parish, Victoria Co., NB in 1901); m. 2nd? Saint John Co., NB 2MAY1915 CLARA CARPENTER. He had a son, JAMES EDWARD HUNT b. McAdam, York Co., NB 24FEB1908, d. Saint John Co., NB 21MAY1938, m. Saint John, Saint John Co., NB 23APR1927 MARY ROSE HUART (HUARD?) (b. Bathurst, Gloucester Co., NB 26NOV1905), d/o Francois ('Frank') and Clara (Couture) Huart. Francois and Clara were m. Bathurst 5SEP1893. (There was a Louis (b. ca. 1846) and Marguerite (b. ca. 1851) Couture in Bathurst in 1881.)
Much of this is guesswork piecing bits together.

Thomas