No, I would not lie. I would empathize about how awful it is to have the flu and offer whatever appropriate assistance you can give.

I believe being genuine is of more value in supporting your friend than having them think you are in the same boat as them. I have no clue what your daughter's friend learnt. When I was in college and taking those 101 level classes, we didn't learn techniques rather a high level overview of the common theories in psychological development.