I agree that there's no (or little) shame in walking a bike up a hill, but I also think that hills need to be looked at as challenges, particularly those you encounter over and over again on routes you repeat, and you need to make a real effort to get to the top of them. There's no better way of getting immediate feedback on your progress as a rider than comparing your performance on the same hill that beat you yesterday. I guess my point is that if you want to improve, don't be afraid to push your limits a little bit on those hills before you decide to hop off.
Years ago I worked with a skinny little guy who was an avid cyclist and weighed about a twig-and-a-half, and when he found out that I had to walk up one particular hill near where I lived, he said "you walk dogs - you don't walk bikes." That little mantra has stuck with me over the years. Even so, it still didn't stop me from bailing on a hill or two at Hilly Hundred the one time I rode it, but not before giving it my all each time.