My aunt lives in my deceased grandparent's home, but is going into a nursing home. Can they take my grandparents home to cover the cost?

Did grandparents have a will? and how was it worded? That Aunt could live there until a time she could no longer care for herself or death? At that time it goes to the surviving children? Or, were one or both grandparents on Medicaid and Aunt allowed to live there but there is a Medicaid lean that needs to be satisfied so the house will need to be sold. If the house was left to the Aunt then she owns it and Medicaid can ask to have it sold as part of her estate. Otherwise, the house is not hers so they can't take it.

Is title to the home still in your grandparent's name, or did they transfer title to (or did your aunt) inherit the home. Who inherited the home after your grandparents died?

Ownership of the home (and how title is vested and recorded) would be an issue. Or was the home willed to your aunt?

In other words, what is the nature of your aunt's legal interest, if any, in the home?

Is her name on the title?

Did she inherit it from them?   Did her parents give it to her...
then..yes, Medicaid will put a lien on it and force the sale.