Can somebody tell me if these sentences are correct?

[michael281001]

1) What Italian food do you like to eat specifically/particularly?

2) I called my aunt and told her to take a good care of her own health before i hang up the phone.

[JMurray]

1) What Italian food do you like to eat specifically/particularly?
2) I called my aunt and told her to take a good care of her own health before i hang up the phone.

michael.
Here's how I would write these sentences:

1) "What Italian food do you particularly like to eat?"
and
"What Italian food do you like to eat, specifically?"

Example 2 is slightly confusing. I'm going to assume that you spoke to your aunt about taking care of her health and then hung up the phone.
But there is another possibility, although unlikely – you may have told your aunt to take care of her health, and to do it while you're still on the phone. To remove this possible ambiguity I would write:

2) "I called my aunt and before hanging up I told her to take good care of her (own) health".

... "own" is probably not necessary, unless, for example, she has been lecturing you about your health and you have told her that it's her health she should be worried about.

not a teacher