"I am a good patient, I always have pain everywhere."

This seems like a typical feature in Slavic languages.

Joining for answer, I also wrote it like that.

There is nothing grammatically wrong with your answer, but it sounds a bit unnatural. You'll be understood, of course but that's not how a native would express this. /copy from "zirkul"/

Fully

You can say "у меня болит везде́"

I am not sure "why" is the right question here, it's languages we are talking about ;-)
Cases of nouns are largely determined by the verbs that take them as objects (and the prepositions that come with them). "Болеть" is a particularly funny verb as it has two meanings and two different conjugation patterns determined by the meaning.
When the subject is a human (or an animal), "болеть" means "to be ill" and it can take an indirect object, the name of the illness, using the instrumental case:
Я болею гриппом (I am ill with the flu); ты болеешь, он болеет; они болеют.
When the subject is a body part, "болеть" means "to hurt" or "to be in pain". For obvious reasons, the first and second person forms are not applicable here but the third person forms are different:
У меня болит спина (мy back hurts); мои ноги болят (my legs hurt).
When used in the latter meaning, "болеть" does not take any objects. I would not consider "у меня" an object, direct or indirect: it's the standard Russian way of indicating possession; it tells us whose body part is in pain.
The past forms are identical for both meanings.

In either case, dative is not the case you can use with this verb, it is always the subject that is ill or hurting.

There is nothing grammatically wrong with your answer, but it sounds a bit unnatural. You'll be understood, of course but that's not how a native would express this.
You can make боль plural however (which is seemingly meaningless since the word is uncountable), then your sentence would sound more natural:
Я хороший пациент, у меня везде боли.

It is acceptable.