I believe as US citizens we can stay in Ireland for 90 days. Is it true, though, if we enter Ireland through the UK we can stay in Ireland as long as six months if the UK stamps our passports for a six month stay?
I may be wrong, but I think you are still subject to the 90 day limit since you are a US citizen. The UK has its rules and Ireland has its rules. The UK can't allow you to stay in Ireland longer than Ireland allows. Just because you enter Ireland through the UK doesn't make you an EU or UK citizen.
I ask because there is now a rule saying those from China and India can do this. I didn't know if other countries are now included in free movement in the common travel area.
When people say Ireland they usually mean the Republic of Ireland which is a part of the EU. Although they are not a part of Schengen, they follow the 90 day limit.
Cross the border into Northern Ireland and you are in the UK where you can visit for 180 days.
As best I can tell from this link -- http://www.citizensinformation.ie/en/moving_country/moving_to_ireland/coming_to_live_in_ireland/visa_requirements_for_entering_ireland.html -- it's far more complicated to do what you described. Go about halfway down the page to "Visa waiver and reciprocal visa arrangement" and follow the links for more info on that. It doesn't seem to be an option for US citizens, probably because we don't have to get special visas to go to Ireland or the UK. Note the list of countries at the end of the page that includes the US. Neither China nor India is on that list.