What do Iraqis find preferable? Saddam Hussein, or a US invasion?
I have the feeling the answer to that is "Saddam Hussein" for most of them. Another way to put it is "better the devil you know than the devil you don't." As Jester said, it would have helped if the US hadn't hung regime changers out to dry in the past; this is well within living memory for the people in Basra, and they know exactly why it happened: in the judgement of the USA in '91, Saddam Hussein was better than a religious Shia government. They probably think that's still the case from the US point of view, and they would simply be changing oppressors from one of the locals (who is at least muslim, if not Shia) to the crusaders from across the sea.
There certainly hasn't been any indication that the US plans on respecting their wishes in governance; to the contrary, the US is making it clear they plan on turning Iraq into a clone of the US.