If I cover my kids under my health insurance as a dependent do I have to claim them on my taxes or can my ex wife claim them on hers ?

One has nothing to do with the other ... so if you are the custodial parent you can allow the ex to claim them by giving her a signed form 8832. 

Per the IRS: 

 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; the custodial parent** is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, 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 exemption to him.

 So, it's good idea to let the other parent know that you will be claiming those items, as many first time divorced parents are not aware of this rule and may try to claim those items, which will cause the IRS to send out letters.

Ref: https://www.irs.gov/publications/p17#en_US_2017_publink1000170897

Scroll down to "Children of divorced or separated parents (or parents who live apart)"

** The IRS goes by physical custody, not legal custody, for who is the custodial parent.  Furthermore, for tax purposes, there is no such thing as joint custody, regardless of what your legal agreement says. The requirement, to be custodial parent, is that the child live with you MORE than 50% of the time. One of you has to be the custodial parent and the other the non-custodial parent. The IRS goes by physical custody, not legal custody.