Common English usage permits so many different positions for the adverb! But I often like it as close to the verb as possible (not always).
So I prefer #1 to #2, although both are clear enough. And, even more than #1, I prefer "She carefully broke the eggs into the bowl", which makes "carefully" yet more important.
One thing you can't say is "She broke carefully the eggs into the bowl" because it's confusing to put "carefully" in between the verb "broke" and its object "the eggs".
If you were giving instructions you could say "Carefully break the eggs into a bowl" if you really wanted to emphasize the care necessary right from the beginning, a good idea when breaking eggs. But you could also say "Break the eggs carefully into a bowl" or even "Break the eggs into a bowl carefully", although this is the weakest.
Isn't English fun! Whoever invented it should be....