Learn Hardcore Hungarian: Vasárnap a család nem együtt, hanem külön eszik. - On Sunday the family does not eat together but separately.

Vasárnap already means on Sunday, so you do not add a separate preposition like English on.
The form vasárnapon exists but is much less common and can sound more formal, old-fashioned, or stylistic. In everyday speech, people almost always say vasárnap.

So vasárnap by itself is correct and normal for on Sunday.

Why do we need the article a in a család? Could I just say Vasárnap család nem együtt…?

You generally cannot omit the article here.

a család = the family (a specific family, usually understood from context, often “our family”)
egy család = a family (some family, not specified)
Bare család without any article is usually wrong in this position in standard Hungarian.

In this sentence we are talking about a definite, known group, so Hungarian uses a család just as English uses the family.

Vasárnap család nem együtt, hanem külön eszik. sounds ungrammatical to native speakers.

Why is the verb eszik singular? In English we say the family eat or the family eats. Should it be plural?

In Hungarian, a család is grammatically singular, so the verb agrees in the singular:

a család eszik – the family eats

Even though the family consists of several people, the noun itself is treated as one unit. English is similar when you say the family eats (American English strongly prefers the singular here).

If you really wanted to emphasize the individual people, you could say:

A családtagok külön esznek. – The family members eat separately.

Here családtagok is a clear plural noun, so you would use esznek (third person plural).

How do you conjugate eszik (to eat)? It looks irregular.

The infinitive is enni (to eat), but the present tense stem is eszik, which is indeed a bit irregular.

Indefinite conjugation (no specific object):

én eszem – I eat
te eszel – you eat (singular informal)
ő eszik – he / she / it eats
mi eszünk – we eat
ti esztek – you eat (plural informal)
ők esznek – they eat

So in the sentence a család … eszik, eszik is third person singular.

Why is nem placed before együtt instead of before eszik? Could I say nem eszik együtt?

nem comes immediately before the element that is being negated or contrasted.

nem együtt, hanem külön eszik – not together, but separately eats
Here, you are negating how they eat (together vs separately).

If you say:

nem eszik együtt, hanem külön eszik

this is also grammatical and clear, and many people would in fact say it this way. The meaning is essentially the same; you just repeat eszik.

The original version (nem együtt, hanem külön eszik) is a bit more compact and keeps the focus directly on együtt vs külön.

What is the difference between nem … hanem and nem … de?

Both involve a kind of contrast, but they are not interchangeable.

nem … hanem = not … but rather / but instead
It corrects or replaces the first option with the second one, after a clear negation.
- Nem együtt, hanem külön eszik.
  They do not eat together, but separately instead.
de = but / however
It contrasts or adds an opposing statement, without necessarily cancelling the first one.
- Együtt eszik a család, de nem beszélgetnek.
  The family eats together, but they do not talk.

In your sentence, because the pattern is “not X, but (rather) Y”, hanem is the correct conjunction.

What exactly is the nuance of külön here? Does it mean “alone”?

külön means separately, not alone.

külön esznek – they are eating separately (e.g. in different rooms, at different times, or at separate tables)
egyedül esznek – they are eating alone (without company)

Also common is:

külön-külön esznek – each separately, one by one (stronger emphasis on each individual)

In your sentence, külön eszik just says that the family does not share a common meal; each person (or subgroup) eats in a separate way or place.

Can együtt and külön move to other positions in the sentence?

Yes, as adverbs they can move, but word order affects emphasis.

Some natural variants:

Vasárnap a család nem együtt eszik, hanem külön.
Vasárnap a család együtt nem eszik, hanem külön. (more marked; emphasis on együtt)
A család vasárnap nem együtt, hanem külön eszik. (focuses more on when, Sunday)

Typically, the focus (the most emphasized contrastive element) stands just before the verb. In the original:

nem együtt, hanem külön eszik

the contrast együtt vs külön is in front of eszik, so that is what is highlighted.

Is the comma before hanem obligatory in Hungarian?

Yes, in standard Hungarian orthography you put a comma before hanem when it connects two clauses or contrasted parts, especially in the nem … hanem … structure.

nem együtt, hanem külön eszik

This is considered the correct punctuation. Without the comma it would look incorrect in formal writing.

Why is there no pronoun like ők (they)? Could I say Vasárnap ők nem együtt, hanem külön esznek?

Hungarian usually drops personal pronouns because the verb ending already shows the person and number.

In your sentence, the subject is explicitly a család, so there is no need for ők.

Vasárnap a család nem együtt, hanem külön eszik. – fully natural.

You could use a pronoun if:

You replace a család with a pronoun:
- Vasárnap nem együtt, hanem külön eszünk. – On Sunday we do not eat together, but separately.
You want to strongly emphasize they as opposed to someone else:
- Vasárnap ők nem együtt esznek, hanem külön, mi viszont együtt eszünk.
  On Sunday *they do not eat together, but separately, whereas we eat together.*

But in the original context, adding ők would be unnecessary and stylistically odd.

Can I change the word order to start with A család instead of Vasárnap? Does it change the meaning?

You can, and the basic meaning stays the same, but the emphasis shifts.

Vasárnap a család nem együtt, hanem külön eszik.
The sentence is about what happens on Sunday.
A család vasárnap nem együtt, hanem külön eszik.
Now the topic is the family, and you say what happens with them on Sunday.

Both are correct. Hungarian word order is flexible, and the first element is usually the topic (what you are talking about). The focus (here the nem együtt, hanem külön part) still stands before the verb eszik, so that contrast remains highlighted in either version.