Hi Heather,
It depends if you have the paperwork to back this up and if you can get each side's story straight. I've had years of extensive medical claims (for infertility and high-risk pregnancies, including twins) and over time there have been mistakes with insurance claims that have had to be sorted out. Sometimes in their favor, sometimes ours. It's never easy.
Your case is unusual, because the HR dept and the insurer don't seem to agree about when coverage started. Why is your husband's company billing you and what is the $1800 for exactly?
If the company clearly made a mistake in changing the status of the policy, part of your children's claims were not covered and it's too late to file a claim, I think they shouldn't charge you if they're at fault. If it was an honest mistake, like underbilling premiums, then I think you should pay (word will get around to co-workers if you don't and that's never good).
If the policy was employee + 1 family and you are the "+1" the birth should have been covered (with you as the patient). It might have clasified as high risk, and the hospital cost could have been far greater than the $1,800 - sometimes tens of thousands.
EOBs for the entire hospital stay (both for you and the babies) will show how much was charged, how much was allowed and paid, and how much you were responsible for (deductible, copay, coinsurance and non-covered amount). Sometimes your responsability can run pretty high, especially with frequent tests done in twin pregnancy - for example $1,000 bloodwork for my pregnancy done in one occassion had $200 coinsurance. A friend's high-risk pregnancy had hospital charges of $60,000 (I'm not sure how much she paid).
That said, you need to contact your insurance company to see when coverage for the children begun, and what your deductible and coinsurance limits are. Write down dates, times and who you spoke with at the insurance company. Confirm what was spoken in writing by fax or e-mail, with copy to your HR. If you have evidence of when you notified HR/ insurance company regarding the birth, all the better
I doubt he would lose his job or coverage if you fight it, just line up your ducks in a row, "gently" gather as much paperwork and evidence both from your employer and from the insurance company, especially with all the EOBs, and try to see where the disconnect happened.
If it's for past due medical expenses - see if there were any bills sent from providers directly at the time - which would happen if there was no insurance coverage. It's possible that one of the many providers present at a twin birth wasn't in-network with your insurer. In any case, if a provider hadn't been paid by now, you would have heard from their collection department a long time ago. If it's a bill that shows up three-years after the fact, there might be a statute of limitations in your state with regards of when you can be charged.
Or as M.R. states, it could have been that your company didn't charge you for family coverage (only employee +1) and you had family coverage, in which case the $1800 is the difference in premiums for these three years (paycheck deductions before/after birth can show this). That you should pay, if not legally, then morally.