so far. No explanations are needed.

On a need-to-know basis and she doesn't need to know other than when the new insurance will start

Will he be able to do Cobra until the new job kicks in?

In our case the new job did not provide health insurance so immediately the girlhippo files to get extra CS to cover the lack of insurance even though her job had Cadillac health insurance that was free to her

Chef went and blabbed to her that he had changed jobs so she went around telling the skids that "dad got fired."

Is BM the type to cause major drama over this?  Take him back to court? Take the kids to the dr/hospital to run up bills to "teach FH a lesson" type?   I might just sit on this info if the kids are healthy and you can ride it out.  Can she get insurance herself for them?   It's so hard to know what to do in this situation with a HCBM,  I remember.  

We opted to pay the full cobra amount than tell BM shit during a slight gap. This is an option whether you quit/laid/get fired. If HR hasn't told you anything you should be getting a packet in the mail. 
 

Cobra let's you keep the current policy you have for 18months I think, assuming you are in the states. If the company was covering a portion they stop and you have to pay the full amount yourself. 

He needs to pay for Cobra. Don't rely on healthy kids. Nephew did and did not tell the ex. It didn't work out as his child had an accident and needed X-rays and treatment. It was a very expensive lesson. Cobra would have been much cheaper. 

Years ago, the job that DH had at the time, technically changed his status to part- time, so could no longer insure SD. His lawyer told him to send a certified letter to BM with the specifics.

As far as the divorce decree requiring him to carry insurance on SD, it stated, "if able to provide through employer".

What you wrote is good but like some others said COBRA or a gap insurance needs to be in place. Not only does he risk being in contempt but what if, God forbid, the kids are in a car accident and there is no insurance. You don't want those bills!! It is worth the price of Cobra to be covered. 

BM also needs to know what the plan is and if she needs to reschedule any appointments so that there are no surprises. 

As wisely stated above, It would be in his best interest to make sure the kids are covered through COBRA during the gap.  If he is responsible for their medical bills and coverage it would be much cheaper to eat the cost for a few months rather than deal with an expensive emergency.  My kids have had silly things happen that would have cost thousands had we not had insurance. Stitches for example, from cutting a finger on a sharp rock at the river. Stitches again from a shard of a broken Corelle plate in the sink. Again...stitches from falling from a chain-link fence. Also, if he is getting a CS discount for providing coverage he could be liable for extra CS if it lapses and makes BM incur extra costs.  Avoid court and extra drama by extending the coverage.  Please let DH be the one to inform her, for one it's his job, also it is annoying and obvious when the stepmom is posing as the husband in emails. 

He  may be eligible for a fully paid COBRA subsidy through ARPA until his new coverage kicks in if he was let go rather than left the company voluntarily.  He should ask his old HR department out it.