I did a show recently where a character dropped a fresh egg, and we had the same dilemma. The director and I did an 'egg call' with the actor where we tried fresh, soft-boiled and hard-boiled, and decided that soft-boiled was the way to go, as the result was still a little bit 'splatty' but didn't leak everywhere.
You might want to try getting a very cheap box of eggs and boiling them for multiples of time, eg 4 mins, 6 mins and so on, to find the degree of 'cookedness' that would work for your show ?
Lizzie