A.,
First, what's done is done in that you chose to have children with someone you knew was not the kind of relationship strong enough for permanency and now the children will suffer for it. You are both right - he is right that it is not good for the children to be tossed back and forth between you two. And you are right - it can be better to come from a broken home than to live in one. But there is a third option - you can heal the broken home!
Since he is their father, he must support them and have a relationship with them. I don't know if you have health insurance or not. But, if you do, please look and see if you can get some couples counseling, or go to counseling alone. Or seek out some free counseling (therapy) services.
Being a single mom is very difficult. I do not wish it upon anyone. Discuss with him the things that bother you. If he wants the relationship, the family, he will need to make some changes. AND - you need to hear him out on the things that bother him and you need to be willing to make some changes.
This may not be what you want to hear or what you want to do. But, it may be better for the children. I don't know your specific circumstance, so it may not be. If not, it means you will have to work full-time even if he does provide child support. That means the children will be in childcare and won't see you as much. You will be tired more and the children won't get as much play time with you - and children need one on one time with you.
So, I suggest you go very slowly in this break up and reconsider it. Both of you be willing to compromise and come to an agreement on how things can be better for you to be a family.
If that won't work after you give it several months trying and you still feel being apart is the best thing, then you will have to fight for child support and he will have to fight for shared parenting time. Let him have it - his relationship with the children is not one you have the right to interfere with.
Be careful what the children see. They are learning from you both about what a family should look like and they are learning from your actions and your behavior. You will all be in my prayers.