What is Chapter 10 of Cheaper By the Dozen about?

Chapter 10 of Cheaper By the Dozen is, for the most part, about health and doctors.  Like most of the chapters in the book, it centers on the eccentricities of Mr. Gilbreth and on his obsession with time management studies.
The main “action” in this chapter has to do with the Gilbreth children getting the measles and also getting their tonsils removed.  One major theme of the chapter is Mr. Gilbreth’s attitude towards his children’s health.  His approach is to simply demand that they not be sick.  At the beginning of the chapter, he is quoted as telling his children

You've been given health, and it's your job to keep it. I don't want any excuses. I want you to stay well.

This is very much in keeping with Mr. Gilbreth’s tendency to be (at least seemingly) domineering and demanding when he deals with his children.
However, the chapter also shows us that Mr. Gilbreth is really not a bad or domineering man, no matter what he tries to pretend.  When the children (almost) all get measles, he paints red spots on his face to try to make it look like he has measles too because he misses being around them.  When he gets his tonsils out, he falls apart (after telling the kids there is nothing to the procedure) and has to be nursed back to health for weeks.  These things show how Mr. Gilbreth likes to bluster and pretend to be very severe even though he really is not like that.  This is one of his eccentricities.
The other major theme of the chapter is Mr. Gilbreth’s fascination with trying to make everything as efficient as possible.  We are told that he believes that he could make surgeons more efficient and has filmed an operation.  He critiques the movements of the physicians, talking about how much more efficiently they could be working.  He even tries to film his children’s tonsillectomies, but is foiled when his cameraman forgets to take the lens cap off. 
Thus, Chapter 10 of Cheaper By the Dozen is largely about health, but it is also meant to emphasize the main topics of the book, which are Mr. Gilbreth’s eccentric ways and his fascination with efficiency.