They definitely need sunscreen as their skin can burn just like ours. Yesterday I was out with the dogs and had Ruthie's hair in pigtails and was concerned about the skin showing where her hair was parted — so I slathered on some sunscreen. We were outside for hours yesterday at Woofstock and she didn't burn. I had read on Google or something that dogs can burn - particularly their nose areas - especially if they have a 'dudley' nose (brown nose or patches of lighter areas on their nose). I would think you would want a special sunscreen for the nose area as they would likely lick it off.
What about a sunscreen spray for your little one since you're wanting to apply it all over? And I agree with jennzy, baby sunscreen might be your best bet. I know they make dog sunscreen too.
Sunscreen for Dogs / Pet Sunblock