We share 50/50 custody, so we each claim a child. He pays both of their health insurance. He is saying that Turbo Tax is asking for it.... because he "received medical tax credit for your dependent". Is this something I need to give him?
No. You do not need to provide him with your social security number, unless he is claiming you as a dependent. Make sure your former spouse is filing as Head of Household when it is his year to claim the children.
In this case it is possible that TurboTax may be asking your former spouse for your social security because he is still filing as "Married Filing Jointly"