The protein in egg white helps texture shaken drinks. Always dry shake if a recipe calls for egg white.
Eggs vary in size so the white from one egg isn’t a particularly accurate measurement. Add the white from a couple eggs to a squeeze bottle and use the same amount every time. Egg whites should keep in the fridge for about a week but don’t add more to the bottle than you think you’ll use and wash the bottle well before refilling.
If you’re worried about drinking raw egg, don’t. But fresh eggs should be fine and high proof booze should kill anything anyway.