Everyone

[ROBINDESBOIS]

Everyone cooks for themselves
Everyone preprared their own food

is the agreement right in these 2 sentences? What´s the rule?

[wrholt]

We can't give you a simple rule for your question. 50 years ago any high school English teacher in the US would have corrected your sentences to:

Everyone cooks for himself.
Everyone prepared his own food.

However, changing social attitudes toward language and sexism mean that using only 'he/his/him/himself' for generic third-person singular reference is rejected by some speakers, and there is no universal acceptance yet for any alternative.

That said, many people favor your versions as the least objectionable alternative.

If you have a specific audicence for which you are writing, you may want to see whether that audience has a strong preference for one alternative over another.

[Rusty]

Both sentences are correct.

Everyone is a singular noun.
Referring to everyone with a plural pronoun, like their or themselves, is proper is British English and allowed in American English, more so in recent years.
In traditional American English grammar, 'his' was used to refer to an indefinite pronoun (which describes 'everyone'). But then folks complained, so 'his or her' was considered more appropriate. Now, 'their' (themselves) is common usage. 'Their' can be, and is very commonly, used to refer to a singular or a plural noun.

Plural antecedent with a plural pronoun:
Students should write their name legibly.

Singular antecedent with all of the possible pronouns:
A student should write his name legibly.
A student should write his or her name legibly.
A student should write their name legibly.

Everyone cooks for themselves.
Everyone cooks for himself.
Everyone cooks for himself or herself.

Everyone prepared their own food.
Everyone prepared his own food.
Everyone prepared his or her own food.

[ROBINDESBOIS]

Both sentences are correct.

Everyone is a singular noun.
Referring to everyone with a plural pronoun, like their or themselves, is proper is British English and allowed in American English, more so in recent years.
In traditional American English grammar, 'his' was used to refer to an indefinite pronoun (which describes 'everyone'). But then folks complained, so 'his or her' was considered more appropriate. Now, 'their' (themselves) is common usage. 'Their' can be, and is very commonly, used to refer to a singular or a plural noun.

Plural antecedent with a plural pronoun:
Students should write their name legibly.

Singular antecedent with all of the possible pronouns:
A student should write his name legibly.
A student should write his or her name legibly.
A student should write their name legibly.

Everyone cooks for themselves.
Everyone cooks for himself.
Everyone cooks for himself or herself.

Everyone prepared their own food.
Everyone prepared his own food.
Everyone prepared his or her own food.
Thanks