hard-boiled can be hyphenated in English
How would you say soft boiled eggs?
Also, scrambled eggs are ביצים טרופות, I had a fun experience in a restaurant in TLV trying to visually describe scrambled eggs a few weeks ago :D
Does this always mean hard-boiled eggs?
I answered with "hard-cooked eggs," which I think of as meaning hard-boiled eggs but which is closer to a literal translation of the Hebrew. It wasn't accepted, but I'm reporting it.
In english: May one say "eggs well done" ?
"Well done" is more appropriate for meat, such as beef, than for eggs, at least in American English. See my comment to Ingeborg above for more info.
"well done" would apply to scrambled eggs or omelettes. One would not say that for boiled eggs.
This is like italian "uova sode" uova=eggs sode=tough, hard, but "boiled" in case of eggs.