My health insurance company has cancelled my policy because they did not receive updated banking information; however they did receive payments, and they are paid to date. They claim since I did not give them our new bank account info before June 27, they are cancelling our policy effective June 30.

If they were paid, that's no basis for cancellation. But the date you mention was June, and this is July. So I am not sure what the issue is, as it is cancelled. Write to the CEO of the Company and ask for an explanation.