Will they be able to qualify for govt insurance - despite their parents' plan?
Are you talking about Medicaid? There's no other government insurance unless your kids happen to be over 65 or are experiencing kidney failure.
And the answer to that depends on various things (do you claim them as dependents on your taxes?). Presumably you're talking about now, in which case the answer is no, they're unlikely to be eligible for Medicaid in your state (the answer will be different in 2014).