... should I bring my air rifle, my .22 - or will I need something more powerful?
It all depends on which variety of egg you plant to hunt.
In the olden days, one could simply hunt eggs with a BB gun. Today’s eggs, however, are of an whole different ovum. They no longer simply lie on the ground waiting to be "picked up".
Cage-free eggs are much more aggressive than the laid-back organic egg. I suggest a .22 for these.
Pasteurized eggs are a bit fried to start with, so you can probably use an air rifle to hunt these.
If you plan on hunting free-range eggs, you might want to use a sniper rifle.
You can use an air rifle on vegetarian eggs, so long as you keep them on the sunny side.
Hunting wild deviled eggs is extremely hard with either a .22 or an air rifle. These are “eggs with attitude”. Some have said that they’re born boiling mad. Even if you do shoot them, they don’t go over easy. These days, nearly all deviled eggs are farm raise as it makes them more coddled.
Make sure you have an egg-hunting license before you go out hunting, as poaching eggs is a felony.