The computer is correct that "to be reunited" is a passive construction because the subject of the verb is not the one causing the reuniting. (See the edit below for a more detailed explanation.) A passive construction, however, does not entail that good style demands recasting the sentence.
It is simple minded to object to the passive as an invariable rule. There are sometimes good reasons to prefer the active voice. There are sometimes good reasons to prefer the passive voice. I do agree that careless writers frequently use the passive voice when the active voice is preferable, but good writers freely use the passive voice when it is appropriate. A simple computer program can highlight use of the passive to permit consideration of whether the active voice is superior in a particular case, but that is no substitute for stylistic judgment in each case.
Style is inevitably personal. My personal rule is to avoid the passive when designating the actor is both possible and important. In your original sentence, I presume that the re-uniting actor will be understood by the audience, and thus I find the passive construction unobjectionable in this case. Nevertheless, the sentence is prolix.
Notice that your re-written sentence conveys less information than your original. If the information omitted will be understood by your audience, that is fine, but in that case I would write "Eligible clients fall into two categories" because that gives the important information clearly and quite concisely. But this is a matter of style, not grammar: there was no grammatical error in your original sentence. Computers are not yet great stylists.
EDIT: As tchrist correctly pointed out, the passive phrase was not passive because the actor was unspecified. That is not the defining characteristic of the passive.
A transitive verb used in the active voice specifies who or what performed the action as the subject of the verb and what the action was performed on as the object of the verb. "Mary hugged Bill" means that Mary, the subject of the verb, was the one doing the hugging, and Bill, the object of the verb, was the one getting the hug. If the verb is in the passive voice, the subject of the verb is what the action was performed on. "Bill was hugged" still means that Bill was the one getting the hug even though "Bill" is the subject of the sentence. Notice in that example the actor is unspecified, which is possible when using the passive. But leaving the actor unspecified is not necessary: "Bill was hugged by Mary" identifies who gave the hug, namely Mary, but "Bill" is still the subject of the sentence.
