Everybody needs someone who loves and cares about them.

You're getting "everyone" and "every one" confused. "Everyone" is a shorter way to say "every one of us." "Every one of us" is more emphatic. "Everyone" is more concise.

probably better say "all of us" or everyone needs everyone of us - грамматически так сказать можно, но для носителя языка это звучит как масло маслянное everyone needs a person who loves them and cares about them если говоришь о человеке в общем без указания пола, лучше использовать they or them даже в единственном числе

Everyone needs someone who loves and care about them.

Each one of us needs a person who loves us and cares about us.