I used to say this. Based on my limited knowledge, it seemed correct. My last name was "La Porte" which was French, and those folks hailed from Quebec - ergo "French Canadian."
My mom's name is "Eagan," very Irish. And, my dad's side had a fair amount of Irish surnames sprinkled about (Blaney, e.g.) - and some English sounding ones (Rock, Jessy, Cox) ergo "mostly Irish." My research to date, though, has shown the inaccuracy of this statement.
Perhaps one of the biggest errors was in assuming that the "Rocks" on my father's side were of English or Irish origins, instead of, as I learned, also French Canadian (originally "Roque" or "Laroque"). It also ignored a lot of what seems to be English blood. Finally, it relied on some pretty serious innumeracy that is exposed when you start looking at pedigrees.
Just looking at the "ethnicity/national origin" of the surname at generational levels (an only rough proxy, admittedly), I now know:
G1 (e.g., Me) - French Canadian ("FC")
FC = 1
G2 (my folks) - LaPorte = FC and Eagan = Irish ("I")
G3 (my grandparents) - La Porte = FC, Rock = FC, Eagan = I, Maloney = I
G4 (GGPs) - LaPorte = FC, Blaney = I, Rock = FC, Rock = FC, Eagan = I, Clarke = English(?) ("E"), Maloney = I, Hanley = I.
FC = 3
I = 4
E = 1
G5 (2GGPs) - LaPorte (FC), Geaudreau (FC), Blaney (I), Cox (E), Rock (FC), Dashnaw (FC), Rock (FC), Jessey (FC), Eagan (I), Tierney (I), Clarke (E), Donnelly (I), Maloney (I), Weed (E), Hanly (I), O'Connell (I).
FC = 6
I = 7
E = 2
At G6, I lose some lines, so its not really measurable.
Mistaking "Rock" for English or Irish, (as with "Goodrow" and "Jessee"), accounted for a pretty significant under counting of the French Canadian heritage. Numerically it seems to be slightly more Irish and especially if Clarke is actually Irish, its certainly not "mostly."