How can I obtain USA Citizenship for my two young children?

To show your physical presence in the United States, you can use any documentation that you may have. For example, you should be able to get social security administration report on your earnings in the U.S. You an also, use things such as your rental receipts (if you rented an apartment, etc.), or if you don't have those, you can maybe get letters from the landlords. If you had a bank account, you can get records of that. If you worked, you should be able to get letters from the employers about your work in NY. If you went to school, the schools would have records of your schooling, etc. You could also provide affidavits from people who could testify that you were in the U.S. during those years.

You will have to try knocking on every door: your parents and you must have left a paper trail of your presence in the U.S.; and tax returns are not nearly the last hope. Your strong hopes are Social Security Administration records and the records of the Border Patrol (which was supposed to keep track of your parents' and yours exits from the U.S. and re-entries). Use Freedom of Information Act to request these records.There might be still available residence leases, insurance records, Registry of Motor Vehicles and voter registration records, newspaper subscriptions, utilities' accounts, public library membership records, etc. Also, do not neglect asking your parents' employers and your own employers to search their personnel records. If you can recall which medical office and/or hospital and which dentist were taking care of your and your parents' health, the medical records can establish your presence in the U.S. just as conclusively as tax returns. Finally, if everything else fails, you can try submitting sworn statements of the people who knew you during your years in the U.S. and can testify to your presence here.

There are many passes to get to the information you need. One of those is through affidavits from people who knew you, worked with you, spent some time with you, and still living in the US. You would have to conduct first investigative services, collect documentation. I believe, you won't be able to do it without a help of a private investigator and then an attorney who will put it all together for you.

To prove residence from 1965 - 1968, you can submit your old passport to show that you left the US in 1968, which implies that you were living in the US prior to that. You can also submit your mother's records, such as tax returns, social security statements, banks accounts, leases, etc. to so that she and you were in living in the US at the time. To prove residence from 1985 - 1987, you can submit your social security statement instead of your tax returns, because social security keeps track of your reported income for future benefits. In addition, you can also submit copies of your bank records, leases, affidavits, driver license record, property titles, etc.