They are not required to agree to any payment plan. However, if they sue you it will take time for the legal papers to be drafted and filed; a summons to be served upon you; a court date and a judgment to be entered. You can consult with a bankruptcy attorney immediately to determine if you are eligible to file bankruptcy and should do so -- most of the time it is a matter of everything starting to go a certain way before the situation becomes such that bankruptcy is the only option.
There are several considerations an attorney should discuss with you. The creditor is not required to work on your terms and often the creditor's preference is to move forward with collection. Enforcing the contract is the creditor's right. There are, however, limits on what the creditor can do and there are options for you to consider.
First, if the matter proceeds to court, you may be able to discuss different terms to pay any judgment that is entered. Also, the interest rate on a judgment is less than the rate you are likely paying the creditor. Also, there are new proposals to adjust the interest rate on judgments in Illinois in certain circumstances.
Regardless the first point, it may be that the terms you are able to afford are less than the creditor will accept. In that event, the creditor is most likely to garnish your wages or try to attach your property (such as money in bank accounts or even personal property). The amount of wages that may be garnished is limited by law and certain of your property is exempt from judgment. So, you may be able to limit or avoid certain efforts at collection. Often a wage garnishment results in a lower payment than the creditor has previously sought, but that still leaves the issue of whether the judgment is actually being satisfied.
I expect in your circumstances that other creditors seeking collection is likely. You phrased your question as "Two of my credit card companies . . ." This suggests there are other debts. You should consult an attorney regarding bankruptcy options, even if you have filed before, even if you own a home or cars. Bankruptcy options may exist to help you on the road to financial recovery.
Finally, debt collectors often overreach. The Fair Debt Collection Practices Act may afford you rights if the collection efforts have violated the law. An attorney can review the documents you have received and discuss the nature of the calls you have received to consider appropriate action.
Your question involves more than can be discussed in the limited space and time of this site and requires an attorney to consider more information from you. I suggest you call an attorney with experience in the above-discussed areas.