Speedboat,
(this is not intended to slight anyone...)
Under the FARs, United is a FLAG carrier and Southwest is not. The CARs don't make a similar distinction. Having the canadian flag on your fuselage doesn't make you a 'designated' canadian FLAG carrier in the same context.
FYI SWA and UAL fleets all have the Stars and Stipes on their fuselage.
If you applied the same criteria in Canada, then Air Canada would be the ONLY FLAG carrier we have.