Isn't that drop experiencing the same pressure as well? If so, what causes the fluid to escape?
that droplet would be experiencing the same pressure from the air pushing inward, but also a pressure pushing outward proportional to the amount the egg has contracted. more pressure pushing out than in means fluid comes out.
Why would the egg contract in the first place?
It's more likely that the "applied pressure" is simply negative...
It's the same as when you squeeze a plastic bottle. If the cap is unscrewed, the liquid will splash out of the opening. The egg is just a bit less flexible than the plastic bottle, so the effect is less visible.