If they are legally married, he can claim the children. For tax purposes, a stepchild is the same as a biological child. They will most likely file a joint return, so she will be claiming them too. She does not need income, on a joint return, if her spouse has income.
The only thing that matters is where the children lived the most time. Everything else (AGI, child support order, amount of support provided, the fact that you were not married) is not relevant. If you are not the custodial parent, you will need form 8332 from her to claim them. The IRS goes by physical custody, not legal custody. There is no such thing as "Equal nights and days through the year" (except maybe in a leap year and that has other consequences). If you both try to claim the kids, it will come down to proving to the IRS that you had physical custody. You don't try to provide proof at tax filing time, that comes later, after the IRS sends out the letters.
This may be helpful in your negotiations with the ex:
There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit (with form 8332); the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status (not applicable if she is married), and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the dependency to him.
