Auguste FRIOUX and Julienne Augustine BOUDREAUX did have son named Eugene:

SOUTHWEST LOUISIANA RECORDS by Rev. Donald J. Hebert:

FRIOUX, Joseph Eugene (Auguste & Julienne BOUDREAUX) b. 31 July 1861 (MC
Ch.: St. Andrew: v. 1, p. 1) [Morgan City]

FRIOU, Joseph Trasimond (Auguste & Julienne BOUDREAUX) b. 23 Oct. 1863 (MC
Ch.:St. Andrew: v. 1, p. 5) [Morgan City]

Below is my transcription of the 1860 US CENSUS record showing
other children in the family of Auguste FRIOUX and his wife, Julienne Augustine BOUDREAUX of Assumption Parish, LOUISIANA.

1860 US CENSUS
Ward 14 (Bayou Boeuf), Alligator P.O., Assumption Parish, LOUISIANA
July 25, 1860
-------------------------------------------
DWELLING 947/FAMILY 917
--------------------------------------------
FRYOUX, Auguste, 42, m, w, cooper, LOUISIANA
FRYOUX, Julienne, 33, f, w, LOUISIANA
FRYOUX, Theodore, 14, m, w, LOUISIANA
FRYOUX, Victor, 12, m, w, LOUISIANA
FRYOUX, Julie, 7, f, w, LOUISIANA
FRYOUX, Marie, 5, f, w, LOUISIANA
FRYOUX, Augustin, 3, m, w, w, LOUISIANA

So as not to confuse one Julienne BOUDREAUX with the other:

Below is my transcription of the 1860 US Census record showing the family of Edouard/Edward VINCENT and Julienne BOUDREAUX of St. Martin, Lafayette, and Vermilion Parish:

1860 US CENSUS
Western District, Abbeville P. O., Vermilion Parish, LOUISIANA
August 6, 1860
--------------------------------------------
DWELLING 513/FAMILY 513
--------------------------------------------
VINCENT, Edward, 40, m, w, farmer, LOUISIANA
VINCENT, Jullien, 39, f, w, LOUISIANA
VINCENT, Elois [Eloi], 15, m, w, LOUISIANA
VINCENT, Eugenie, 14, f, w, LOUISIANA
VINCENT, Celeste, 11, f, w, LOUISIANA
VINCENT, Elmire, 9, f, w, LOUISIANA
VINCENT, Belsire, 7, f, w, LOUISIANA
VINCENT, Moses, 4, m, w, LOUISIANA
VINCENT, Cora, 2, f, w, LOUISIANA

Census Source:
Ancestry Library Edition