Greg,
Short answer - Don't know. The flag for La Reine is also shown with the fleur-de-lis and crowns on the French vexilollogie site hosted by Arnaud Bunel, and dated at 1661 on that site.
http://www.drapeaux.org/Accueil.htm
(select France, Ancien Régime, Infanterie, La Reine).
The literal translation of Regiment de La Reine would be the Queen's regiment, so the fleur-de-lis and crowns could designate royal status or affiliation. Just an assumption on my part. I've only ever seen La Reine's flag with the fleur-de-lis and crowns except for the one site you linked to. One possible thought is that the flag without fleur-de-lis and crowns is only shown on the "French in Canada" page, La Reine had 3 battalions, perhaps this flag was sent to Canada with the 3rd battalion?
Here is Royalfig's version:
http://translate.google.com/translate?h ... md%3Dimvns
And here is Ian's from Warflag, which also uses slightly different colors in the cantons:
http://www.warflag.com/flags/wss/wssfr4.shtml
I checked Fouré (translated by Condray) and he does not show La Reine in his two volumes. I don't have access to Robert hall's work, but I also checked Mark Allen's WI article, to wit:
Issue 49, page 17 (pics) and 18 - La Reine is shown with the fleur-de-lis and crowns, cantons black and green, drummer in red coat with blue cuffs and Queen's livery (supports the assumption above).
I would feel comfortable using the version with fleur-de-lis and crowns, I think there is enough provenance.
Bill