No. If a debt collector makes such a threat it is only because he has no other cheap way to collect the debt. The creditor may have already reported you to the credit bureau. Debt collectors only make threats like this in order to scare you into paying a debt that is difficult to collect otherwise. Do not let a creditor or debt collector scare you into paying less important debts first!
Should I pay a debt because I have co-signed on it?
Co-signed debts should be treated like your other debts. If you put up important property as collateral, treat it as a high-priority debt. If you didn't, treat it as a low-priority debt.
Should I take on new debt to repay my old debt?
It may seem that the best way to catch up on your bills and debts is to take on new debt in order to repay old debt. However, this can lead to long-term problems and is usually not a good idea. The best way to deal with too much debt is to decide which debts to pay first, which debts to pay last, and which debts to not pay at all.
However, if you decide that you must take on new debt to repay old debt, you should not turn unsecured debt into secured debt. For example, do not pay off your credit card debt with a home equity loan even if it lowers your monthly payments. If you can not pay your credit card debt, the creditor will have to sue you in order to collect any money from you. Even if they do sue you and win, they may not be able to collect any money or property from you. However, if you can not pay your home equity loan, you may lose your house and all of the money you have paid toward its purchase.