Learn Hardcore Swedish: Hennes barn äter ägg och grönsaker till frukost. - Her children eat eggs and vegetables for breakfast.

hennes means “her/hers” and is used when the possessor is not the grammatical subject of the same clause (or when there is no prior subject to refer back to). It never changes form.
sin/sitt/sina is reflexive and points back to the subject of the same clause (sin = common-gender singular, sitt = neuter singular, sina = plural).

Examples:

Hennes barn äter … (Her child/children eat …) — correct at clause start; there’s no earlier subject to refer to, so you can’t use reflexive.
Hon säger att sina barn äter … (She says that her own children eat …).
Hon säger att hennes barn äter … (She says that her children eat …) usually means someone else’s children (not her own).

Is barn here singular or plural?

It’s ambiguous. barn (neuter) has the same form in indefinite singular and indefinite plural:

singular: ett barn
plural: (no ending) barn
definite singular: barnet
definite plural: barnen

With a possessive like hennes, you don’t add the definite ending, so hennes barn can mean either “her child” or “her children.” Context disambiguates.

How can I make it clear that it’s “her child” vs “her children”?

Add an adjective, which shows number:
- singular: hennes lilla barn (her little child)
- plural: hennes små barn (her little children)
Or use quantity words:
- hennes enda barn (her only child)
- alla hennes barn (all her children)
- ett av hennes barn (one of her children) You cannot say “hennes barnet.” Possessives don’t combine with the definite suffix.

Why is there no article before ägg and grönsaker?

grönsaker is an indefinite plural (“vegetables”), and Swedish normally has no article in the indefinite plural.
ägg is one of those neuter nouns whose indefinite plural is identical to the singular form. Without a number, ägg here is understood as plural (“eggs”) or a general/partitive idea. If you mean one egg, say ett ägg.

Why is it till frukost and not för frukost?

Swedish idiomatically uses till with meals to mean “for (a meal)”:

till frukost / till lunch / till middag

för frukost is not used. If you mean “before breakfast,” say före frukost. To refer to a specific breakfast event (like at a hotel buffet), you can use the definite: Vad serveras till frukosten?

Is the word order SVO here? Could I front the time phrase?

Yes. The main clause shows normal SVO:

Subject: Hennes barn
Verb: äter
Objects: ägg och grönsaker
Adverbial: till frukost

You can front the adverbial and keep Swedish verb-second (V2):

Till frukost äter hennes barn ägg och grönsaker.

Could I say Hennes barn äter frukost med ägg och grönsaker?

It’s possible, but the most idiomatic way to name what you consume at a meal is till: äta X till frukost.

äta frukost med ägg och grönsaker can sound like “they have breakfast that includes eggs and vegetables,” which is fine, but äta ägg och grönsaker till frukost is the default pattern.

How do I form the plural and definite forms for grönsak?

singular indefinite: en grönsak (a vegetable)
plural indefinite: grönsaker (vegetables)
plural definite: grönsakerna (the vegetables)

The word is a compound: grön (green) + sak (thing), and its plural takes -er.

What are the principal forms of the verb äta?

infinitive: äta (to eat)
present: äter (eat/eats) — same form for all persons and numbers
past (preterite): åt (ate)
supine (used with har/hade): ätit (eaten)
imperative: ät! (eat!)

Why is ägg the same in singular and plural? How do I show number?

It’s one of several neuter nouns with zero plural in the indefinite:

singular: ett ägg
plural: ägg
definite singular: ägget
definite plural: äggen

You show number with determiners or numerals:

ett ägg, två ägg, några ägg (some eggs), inga ägg (no eggs).

How do I pronounce the special vowels here (ä, ö, u)?

Approximate guides for English speakers:

ä (as in äter, ägg): like the vowel in “bed,” but a bit tenser/longer when stressed.
ö (as in grönsaker): like British English “bird” or “fur,” but round your lips.
u (as in frukost): a front rounded sound; try saying “ee” while rounding your lips, short here.

Stress falls roughly as: HÉN-nes BARN É-ter ÄGG och GRÖN-sak-er till FRÚ-kost.

Where would negation (inte) go?

Place inte after the finite verb:

Hennes barn äter inte ägg och grönsaker till frukost. If you want to negate only one part, move inte accordingly:
Not eggs (but maybe something else): Hennes barn äter inte ägg till frukost.

How do I make a yes–no question from this sentence?

Invert to verb-first:

Äter hennes barn ägg och grönsaker till frukost? Short answers:
Ja, det gör de. (Yes, they do.)
Nej, det gör de inte. (No, they don’t.)