Can a Canadian citizen living in the US with their US citizen spouse apply for Canadian immigration for their spouse and does the spouse have to give up their US citizenship?

Yes s/he can sponser a spouse. For the method please refer to: http://www.cic.gc.ca/english/information/applications/fc.asp As far as giving up citizenship: it is not required that the spouse give up citizenship. The US basically considers the spouse a US citizen and if they have citizenship in Another Country that is irrelevant to the US.