Can a nursing home take a home to pay bills?

It's usually not the nursing home itself which takes a home--it is Medicaid. After your mother stops paying for the nursing home herself, Medicaid starts paying. But Medicaid does not let you keep valuable assets, like a home, while making the taxpayers (who fund Medicaid) pay for you. So Medicaid can take those valuable assets, including homes, to pay for the cost of care.
Be careful about what you do: there is a "5-year lookback" period in the law for this. That means  that Medicaid can "look back" up to 5 years and undo any transfers or transactions made during that time if they were not for fair market value--this is to specifically prevent people from sheltering assets by putting them in their children's, spouse's, siblings' or friends', etc. names. So if you buy the house from your mother, for more or less its current market price, that is a valid transaction: your mother will have to spend the money on her care, but Medicaid won't take the house. But if you just transfer it for free or a nominal amount to the childrens' names, then if your mother goes into a nursing home and Medicaid starts paying for her within 5 years, Medicaid can potentially take the home.
Medicaid law is complicated; you are strongly advised to speak with an estate planning attorney to see what, if any, options you have.