Even though I say the Civic is better, since you're going to get a b18b engine anyway you might as well go for the integra. You're limited to $10000, and an engine swap is not cheap, unless you're willing to do it all yourself. That is not something you can learn on the go very easily. If you bought the Civic you'd end up buying 2 engines. I mean, how easy is it to sell off you Civic engine once you get the b18b in there? Didn't someone mention that turbo isn't great in autocross since you don't have time for it to spool up?
A no-tec is easier to boost, but, if you're good at it any vtec engine will give you more power in the end.