To determine if your children are U.S. citizens we'd need a lot more information.U.S. Citizenship is gained by either naturalization, birth in the U.S. or automatic acquisition. So one would need to analyze if your child born in the Ukraine obtained US citizenship through automatic acquisition, for which usually at least one of the parents has to be a U.S. citizen. Since you state that you have a green card but are not a U.S. citizen, is the father a U.S. citizen? You should in any event consult an immigration attorney directly - also, to talk frankly about the length of stay abroad for a green card holder as you do not want to jeopardize either your current green card or your possible future naturalization by staying abroad for too long.
This advice does not form an attorney-client relationship and is merely informative. It should not by itself be relied upon to address a legal concern.
If you are going for more than 6 months you should consider applying for a Reentry Permit first. You also need to maintain sufficient ties to the U.S., pay U.S. taxes, etc. so that you will not be considered to have abandoned your green card. Any travel greater than 6 months may also have ramifications on when you can apply for U.S. citizenship so I suggest a consultation on these issues before you leave.
Your children are US citizens. They can stay as long as they want. You, to be safe, should return in less than six months. If you have to stay longer than six months, contact an immigration attorney before you leave. Good Luck.