Drywall – eating the drywall

Since the Pest Controller didn't find any pest cause or problem, very good news. Go ahead & patch, tape & mud it. Make sure the patch doesn't touch the masonry. The moisture down there may get it again or any paper break. If it starts happening all over, eventually, then painted fiberglass faced drywall would last or you might opt for cement board at that time. But, just patch what you've got to gauge what works & what doesn't & for how long.

Insects are incapable of "eating" concrete, but they do tend to exploit weak or damaged concrete to access something more appealing, such as a food source or nesting site.

It only takes a small crack in an otherwise secure concrete foundation to let termites in. Poured concrete is the most reliable, but if you have a block foundation, the mortar is likely to provide open spaces. Expansion joints between home sections are another invitation to pests, as is any area where pipes or wires enter the house.

Once inside, termites build dirt tunnels. These tunnels get wider over time, so even though the bugs aren’t eating your foundation, they can cause a great deal of damage as pressure builds up and widens existing cracks. This slow process may go unnoticed until it becomes serious, making regular inspection of your home and foundation an important part of pest control.

Source

Eradication is best left to a professional for something this serious.