Can someone pushing their fingernail into your spine damage the nerves?

No such maneuver will reach the nerves buried under layers of skin and tissue. Pressure over some pressure points can cause some transient interference with nerve function occasionally. Think about what happens when you hit your "funny bone" which is really a nerve.

The fingernail might penetrate the skin, causing bleeding and even eventual local inflammation/ infection, but would never cause vertebral, spinal cord, or nerve damage.