What you Need
What to do
First try protecting the egg using just a strong box.
Cut the bottle open around the widest part.
Put an egg in the bottle and tape the bottle back up.
Try dropping it from about waist height.
Now try filling the bottle with salty water. Keep adding salt until the egg starts to float.
Try dropping the bottle from waist height. Do you get the same result?
What may happen
The egg just in the bottle will smash, but in the salty water it should mostly survive.
Why does it happen?
If you drop an egg on the floor its speed changes from about 3m/s to stopped in a very short period of time, this means that is accelerates (decelerates) very rapidly. A large acceleration requires a large force and because this force is concentrated in one place the eggshell isn't strong enough to support this force and it breaks.
If you put the egg in a bottle the egg still stops rapidly and the force is still concentrated in one place so the egg still breaks.
However if the egg is floating in a fluid then the force that slows down the egg is upthrust, the same force which was making it float in the first place, which gets larger the larger the acceleration of the bottle.
This is one of the reasons that an unborn baby is surrounded by salty amniotic fluid because the baby is a similar density to the amniotic fluid it is floating, so if there are any impacts to the mother the forces are evenly transferred to the baby not doing it any damage.