Hjúkrunarfræðingurinn segir að ég eigi að hvíla höfuðið.

It means the nurse.

It is a compound noun:

• hjúkrun = nursing
• fræðingur = specialist / trained professional

So hjúkrunarfræðingur is the standard word for nurse, and hjúkrunarfræðingurinn means the nurse.

Long compounds are very normal in Icelandic, so this kind of word is not unusual.

Because Icelandic usually adds the definite article to the end of the noun.

So:

• hjúkrunarfræðingur = a nurse
• hjúkrunarfræðingurinn = the nurse

That final -inn is the suffixed definite article here.

Segir is the present tense, 3rd person singular of segja.

• segja = to say
• segir = says / is saying

So Hjúkrunarfræðingurinn segir... means The nurse says...

Here að means that and introduces a subordinate clause.

So:

• segir að... = says that...

This is very common in Icelandic, just like in English.

Because after the conjunction að, Icelandic uses normal subordinate-clause word order.

So:

• að ég eigi... = that I ...

In a subordinate clause, the subject typically comes before the finite verb:

• að ég eigi að hvíla höfuðið

That is the normal structure here.

Because eigi is the subjunctive form of eiga, and Icelandic often uses the subjunctive in reported speech after verbs like segja.

Compare:

• Ég á að hvíla höfuðið. = I should / am supposed to rest my head.
• Hjúkrunarfræðingurinn segir að ég eigi að hvíla höfuðið. = The nurse says that I should rest my head.

So:

• á = indicative
• eigi = subjunctive

A learner will often see this pattern after segir að, heldur að, vill að, and similar structures.

Here eiga að does not mean possession.

It means something like:

• should
• ought to
• am supposed to

So:

• ég á bók = I have a book
• ég á að fara = I should / am supposed to go

In your sentence, ég eigi að hvíla höfuðið means that I should rest my head.

They are doing two different jobs.

1. The first að is a conjunction:

   • segir að... = says that...

2. The second að belongs to the expression eiga að + infinitive:

   • eigi að hvíla = should rest

So even though both words are spelled að, they are not the same grammatical item in this sentence.

Because hvíla and hvílast are different constructions.

• hvíla can be transitive: rest something
• hvílast means rest oneself

So:

• hvíla höfuðið = rest the head / rest my head
• hvílast = rest

For example:

• Ég þarf að hvílast. = I need to rest.
• Ég þarf að hvíla höfuðið. = I need to rest my head.

Because Icelandic very often uses the definite form with body parts when the owner is obvious from context.

So:

• hvíla höfuðið literally looks like rest the head
• but in natural English it is usually rest my head

This is extremely common. Icelandic does not always use a possessive where English does.

If you said höfuð mitt, that would sound more explicit or contrastive, something like my head in a specially emphasized way.

It is the direct object of hvíla, so it is in the accusative.

The noun is:

• höfuð = head (neuter)

Definite singular:

• höfuðið

For this neuter noun, the nominative and accusative definite singular happen to look the same, so you do not see a visible difference here. But grammatically, it is the object of the verb.

Formally, it is the head.

But in this context, the natural meaning is my head, because the sentence already makes it clear whose head is meant.

So a good way to think about it is:

• Icelandic form: the head
• natural English meaning: my head

This is one of those places where a very literal translation can sound strange in English.

Yes. That would be the direct version:

• Ég á að hvíla höfuðið. = I should / am supposed to rest my head.

Your full sentence simply embeds that idea inside reported speech:

• Hjúkrunarfræðingurinn segir að ég eigi að hvíla höfuðið.

So the sentence is built from:

1. Hjúkrunarfræðingurinn segir...
2. ...að ég eigi að hvíla höfuðið.

Yes, and that is a useful comparison.

• Hjúkrunarfræðingurinn segir að ég eigi að hvíla höfuðið. = The nurse says that I should rest my head.

• Hjúkrunarfræðingurinn segir mér að hvíla höfuðið. = The nurse tells me to rest my head.

The meaning is similar, but the structure is different:

• segir að... = says that...
• segir mér að... = tells me to...

Both are good Icelandic, but they are not exactly the same sentence type.