As far as the IRS is concerned, either of you can claim the child, as a dependent, regardless of who's insurance he is covered by. You can decide between you who will claim him (this assumes you live together). Only if you can't agree does the IRS have "tie breaker" rules
The 2nd issue is based on the rules of the insurance plan. Does the employer's plan allow a child that is is not a dependent to be covered. Most, if not all, probably do. Per TomD8's answer, the Affordable Care Act (ACA) apparently requires that.
