Why is "You are to eat" wrong? It's not a common way to say it in english, but it is correct english.
I did that too and it was not accepted. I think it should be, even though it is not the most natural way to say it.
To the best of my knowledge, modal verbs do not indeed use the infinitive particle.
That is understatement XD.... In fact, modal verbs do not use "zu", unless they are used in infinitive constructions, like "Ich glaube fliegen zu können". With auxiliary verbs it is possible: "Ihr habt zu essen!" (i.e. "You have to eat!")
Not exactly like in English and not as often, but it means the same. The negation is different, "Ihr habt nicht zu essen" means you must not eat. Either way it sounds pretty harsh.
It's been my understanding that in the infinitive form of verbs are "to run" or "to eat" already inherently, at least in english. As such, to say "ihr sollt zu essen" to me would appear as if you were saying: "you are supposed to to eat."
That's how it makes sense to me at least so I hope that's helpful.