Does the tsa allow iguanas to come with you on a plane?

I want to bring my fiance an iguana from florida to Canada.It is legal to bring it over,but I would rather not risk shipping it since it will be so cold.I would much rather just bring it on the plane im going on to bring it to him.I dont care if the airline allows it;its small enough so that they would never see it and id keep it in the same shipping box i would if i sent it so it cant get out or anything,I just need to know if I would be allowed to bring it ped airport security.how does that work with animals anyways?ive never brought a dog or cat with me on a flight either.if it's legal to bring then why would the tsa people stop me when they see an iguana in a box in my bag and try to take it away from me?what if I have the box out and clearly marked?