I don't think this is true. I worked with 5 boys and 5 girls to manufacture situations where it wasn't true. I want to avoid writing out a counterexample with 50 people!25 boys and 25 girls sit around a table. Is it always possible to find a person both of whose neighbours are girls?
Is there a way to approach this using permutations? It seems to be that if you put them in Boy/Girl order, and then switch the two people before the person with two girls as neighbours then you get a nice counterexample.
Swapping people around seems like it can be done using permutations.
Anyway, I hope someone can help.