It means the eggs were soft. Rather than break, they "gave way" when they were squeezed. When you squeeze the egg, there are two possibilities. One possibility is that it remains firm, resisting the force of your squeeze, until it finally breaks. The other possibility is that it gives way (surrenders) to your pressure, bending a little bit.

Gave in this sentence matches the following definition: alter in shape under pressure rather than resist or break.

Since "gave" is the past form of "give", this sentence is ambiguous, at least from my point of view. Maybe someone else would have a better answer. 🤔