No, there was no implied contract. You should have worked something out with the hospital finance department; they may have reduced your bill and may have provided you with fairly generous payment terms (not as low as $10/month, though). Your $10/month most likely did not even cover interest.
And no, you must deal with the collection agency now. Most likely, the hospital sold all rights to your account for pennies to the dollar to the agency; the hospital will not be able to accept payments for your account.
Under federal and California law, you could demand that the collection agency stop contacting you to collect the debt. But it will still have the option of filing a lawsuit against you and asking for a garnishment order where you will have to pay up to 25% of your disposable earnings toward the debt.
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No, you can't. There was no "implied contract." You admit you owe the money, and the hospital has every right to collect the debt, including sending it to collections. They can add the collection costs to the bill. You need to deal with this, including the collection agency.
This answer is not to be considered a response to a specific legal issue in a specific jurisdiction - it is to be considered only a general response to a hypothetical scenario posed by the questioner. For specific legal advice, please consult with a licensed attorney in your jurisdiction.
The amount of the charges is the "reasonable value" of the services rendered. This may not be the same amount as the collection agency claims you owe. Also, there may be errors in the computation of the billings. The hospital's itemization of its charges should be reviewed.
You can, within 30 days of your first contact by the collection agency, inform them that the debt is "disputed." You can dispute the amount of the charges, i.e., that the hospital charged more than was reasonable, or that the itemization contains charges for services or supplies that you did not receive. For example, did the hospital charge you $20 for one aspirin pill? If you dispute, in writing, the debt, and if the collection agency reports the debt to a credit reporting agency (TransUnion, etc.) the collection agency is supposed to also inform the credit reporting agency that the bill is disputed.
The statute of limitations is probably 2 years from your last payment. So, each payment you make extends the statute of limitations.
You can ask, in writing, that the collection agency to not call you. Then, they can send you one last letter, telling you that they will not contact you. And, they can file a lawsuit. That will be a considerable expense to them. More often than not, they do not want to invest their money in a lawsuit unless they are sure that they will recover something.
If the collection agency makes a demand for even one cent more than is actually owing, it is in violation of the Fair Debt Collection Practices Act. The collection agency cannot add its "collection charges" to the demand. Interest cannot exceed the legal rate.
If the collection agency sues you, and if you file an Answer / General Denial, the collection agency will need to bring someone from the hospital to court to testify as a witness in order to prove their case. If the case goes that far, you will be in a very strong negotiating position.