This is probably not what the exercise wants, but I would make a distinction between does love and can love. That is, a person may be capable of loving someone, but simply doesn't. Let's write this as $Does(x,y)$ if $x$ does love $y$ and $Can(x,y)$ if $x$ can love $y$. Then the sentence becomes
$$\forall x(\lnot Does(x,x)\implies \forall y(\lnot(y=x)\implies\lnot Can(x,y)))$$
or, if you need to explicitly limit yourself to the universe of people,
$$\forall x(Person(x)\implies(\lnot Does(x,x)\implies\forall y(Person(y)\land\lnot(y=x)\implies\lnot Can(x,y))))$$
Note, the "$\lnot(y=x)$" here stems from the "else" in "anyone else." The sentence as stated does not assert that a person who doesn't love himself cannot love himself; if that were true, psychotherapists would have to admit defeat with a lot of their patients.