glue 4 wheels to a boat and have siit in a trench (road) the bucket floods the trench pushing the egg carrying 'car' to the reactor.
Hmm..an out of the box idea but how do you transfer the soft egg into the 3" deep protective structure with it breaking?
and the vehicle must work on its own so a trench or road would not be allowed.
I don't think an source of potential energy could be used, like a rubber band.
A wheeled vehicle with a paddle wheel of sorts driven by the water, that has a string that would wind up to limit it'* travel and trip a device that would unwind the egg support to gently lower it at the precise destiination point, might work. Much experimentation would be required. Teamwork is always the key.