Yes, it’s “read” all over, not “red”. However, a sunburnt zebra is black and white and red all over.

It's a pun that only works when you hear it aloud. "Read" and "red" are perfect homophones. A newspaper is "read." And--before about 1970 anyway--newspapers only printed photos in black and white. The context, "black and white" tricks you into perceiving the word "read" as "red." The detail "all over" is needed to make the riddle puzzling. It is a little unnatural to say a newspaper is read "all over," and of course people don't usually read ALL of a newspaper. Here's another one, which again only works when spoken aloud: "Railroad crossing, look out for the cars! Can you spell 'that' without any R's?" The answer is "T-H-A-T." The target thinks the word 'that' is a pronoun, referring to the phrase "Railroad crossing, look out for the cars," which obviously contains five R's and can't be spelled without them. The trick answer is that the first line is distraction, and the question is whether you can spell the four-letter word "THAT" without any R's.

Read ( past tense) sounds the same as red ( color). It means you read a newspaper from the beginning to the end.