My children were both born on US soil. My wife is a US citizen, born in the US. I am not. Although, I am a citizen of a very friendly country. ;-) Can my children become president?
Their status is in doubt. Does your nation of birth extend citizenship to your children, or even grant it automatically? Have they accepted this citizenship if offered? If so, then they’re not. They’re eligible for everything else, with the exception of being in the direct line of Presidential succession.
No. Not Constitutionally. They are not natural born citizens if you were not a citizen when they were born. Sorry, them’s the breaks.