But with the OP fixating on the first go only, without giving context as to whether they all pick their egg at once, or take turns to pick, then it's simply about going first. In which case you have the best selection as there are 5 "good" eggs.
That being said, I think the OP is trolling as he hasn't given context yet, or even the name of the video.
The context is the statement 'if I go first, then I have a better chance'
IF you assume that the game will last all 6 turns, and the bad egg is last, this is true but also redundant as you pre determined the outcome.
However, if you are to assess your odds properly, the position you choose to go will confer no advantage to your chances of losing, as per the below.
First round
Chance of getting the egg - 1 in 6
Chance of having to pick - 6 in 6
Combined overall chance = 16.7%
Second round
Chance of getting the egg - 1 in 5
Chance of having to pick - 5 in 6
Combined overall chance = 16.7%
Third round
Chance of getting the egg - 1 in 4
Chance of having to pick - 4 in 6
Combined overall chance = 16.7%
Fourth round
Chance of getting the egg - 1 in 3
Chance of having to pick - 3 in 6
Combined overall chance = 16.7%
Fifth round
Chance of getting the egg - 1 in 2
Chance of having to pick - 2 in 6
Combined overall chance = 16.7%
Sixth round
Chance of getting the egg - 1 in 1
Chance of having to pick - 1 in 6
Combined overall chance = 16.7%
You can't ignore the chances associated with actually taking a turn.