The dedication makes me think that it was some little girl, who in her playtime with her dolls, came up with Dr. Squash, the doll doctor. And the story reads very much like the way a child would play.
The dolls have all sorts of ailments, (the teddy bear gets a nose bleed, the fireman doll needs a splint, the Indian doll fell off his horse into poison ivy, and the poor clown doll has 4 contagious diseases!). Each time he's called, Doctor Squash comes running and gives them the appropriate treatment. Then when Doctor Squash himself falls ill, the dolls all do their best to take care of him.
We just love sweet and winsome books like this! (No surprise it's by Margaret Wise Brown). If J.P. Miller's illustrations look familiar, it's because he worked for Disney and illustrated many Little Golden Books. Here's a really interesting bit about him!