Translate the following expression to English
$$∃x∀y∀z ( ( F(x,y) ∧ F(x,z) ∧ F(y≠z) ) → ¬F(y,z) )$$
So, I have two answers:
There is a person who does not have two friends that are friends of one another.
There is person who has two friends that are not friends each other.
Are they both right?
I'll assume that $F(x \neq y)$ is actually meant to be $x \neq y$.
Talking about 'friends' implies the symmetric property; if $x$ is a friend of $y$ then $y$ is a friend of $x$ too, that is: $\forall x \forall y F(x,y) \leftrightarrow F(y,x)$, which you don't know if it holds. Try instead to interpret $F(x,y)$ using the loving relation: '$x$ loves $y$' (but sadly $y$ may not love $x$...).
Under the later interpretation, the proposition can be paraphrased as 'there is a person $x$ who either loves only him/her-self, or all the people that he/she loves (if there exist at least two of them) do not love each other'.
Tip: In order to avoid assuming extra properties, you should always interpret predicates like $P(x,y)$ as a basic subject-verb-object English sentence (somebody does something to someone), that's what predicates are meant for, after all!