“I am writing to you from Ireland. We moved back here eight years ago – my husband and I and our American citizen infant. We have since had two more children. I had a green card when we left and I guess that it would be no good now since we live here and I haven’t been back since. My husband also had a green card but was in the last steps of applying for U.S. citizenship, so he kept with the application and was approved four months after we returned home. I am wondering, because of that, if the children we have had in Ireland could also have entitlement to U.S. citizenship. Is my claim to citizenship hopeless? We are not planning on moving back to the U.S. These are just questions that I have had on my mind.”
As far as your claim to U.S. citizenship, right now it is non-existent. If you ever decided to return here your husband, as an American citizen, would be entitled to sponsor you for permanent residency. Once you accumulated three years of residence as a green card holder, you would be able to apply for citizenship in your own right.
You are correct in saying that your old green card is no longer of any use to you, given the amount of time that you’ve been out of the country. 
The news is considerably brighter for your Irish-born children. It’s quite likely that the children can immediately stake a claim to U.S. citizenship because of the following rule with regards to its transmission – for children born on or after November 13, 1986, they are U.S. citizens at birth if they have one citizen parent who accumulated at least five years of physical presence in this country, with two of these years taking place after the parent’s 14th birthday. 
It’s 99.9% likely that your husband meets the above requirement, so your children, and any you may have in the future, can immediately formalize their citizenship. This can be done at a U.S. consular office abroad – in your case, the American Embassy in Dublin.
The process is relatively easy, and it would be wise for you and your husband to do it, simply because an American passport is a valuable document that they may well wish to avail of in the future.
You can apply for a Consular Report of Birth Abroad, the first step in obtaining a passport and Social Security number for the children. Further instructions are available at the embassy’s website at