what does a very red boy mean here? a boy with red hair, or a boy with red clothes, or a native American boy? or something else?

It is taking place in a home for old people. this red boy brings the tea and then he comes back to remind the woman about a booster injection.

the author doesn't talk about this boy anywhere else in the book.

I would take it as a temporary condition, so either red from embarrassment or red from heavy exertion like running.  However, it might actually be that the author means a sunburned teen with red hair.  I would expect (and could be disappointed) for other statements to clarify this for me.  What does the author actually mean?  He isn't very clear, is he?  Red is not an expected condition, so what is it supposed to mean?  Ought to tell us.

embarrassed ?  sun burned ?? {we need context}

Red-faced. Angry or embarrassed, as someone already said.

Perhaps a very embarassed or extremely angry boy.