Ok, so I posted this on tumblr earlier and figured I should post here too that after 13 days of caring for my mothra (had the fairy adult) it changed like last time into Gohgo. Only difference between the first time I got the character and this time is I did not raise an adult before this one as I had it tabbed most of last month.
This makes me think that you do not need a special egg to get the secret characters on the mothra but won't know for sure until gohgo leaves in a week or two. I will post again if my theory is right and I manage to get a different secret character with my next adult.
Ok I can now say the whole special egg for the mothra is rubbish!
See 2 days ago I raised Makora on my mothra to see if you needed a special egg or not to get a secret character and indeed you do not.
As you can see last night he changed into Godzilla!