Well I have never participated in Putnam or any other college-level competition, but since no one else has offered their advice I thought I might as well give you my thoughts.
You will probably be hard-pressed to find a book that deals exclusively with putnam-level single-variable calculus problems. A good preparation is any of the standard rigorous books (Apostol, Courant, Spivak) coupled with standard rigorous real-analysis books (I only have experience with Rudin's PMA, but that seems fine to me). If you truly understand all that stuff and have done some general problem-solving you should be able to tackle hard problems and with enough practice Putnam problems. As for actual books; I have Berkely Problems in Mathematics which is a pretty neat collection of problems. They are not all calculus and I guess most is below Putnam-level, but still good training. For high-school-level competitions I have good experience with the stuff by Titu Andreescu and Kiran Kedlaya and while I haven't read any of their college-level books I guess the books "Putnam and beyond" and "The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions, and Commentary" would be of use to you (again not exclusively calculus, but neither is the Putnam).
Learn the theory and then practice a LOT of problems is my guess.