In order to get ur eggs u have to do all the parts to it. Orphic can be get I think after the 2nd trade. Unsure. This is how I did my eggs.
Initial 3 -
My name is Bonkerskitty, so I had to get B-O-N. And trade it to the moogle.
Reward -
I got 2 Chocobo Tickets
Initial Straight 8 -
The first 8 laters after the first letter of ur name according to the alphabet. B-C-D-E-F-G-H-I
Reward -
I got my Fortune Egg
7 of a Kind -
7 Letters all the same.
Reward -
I got the Happy Egg.
Orphic Egg -
Trade the moogle the first 5 letters of the current day. IE. I did mine on Earthsday, so I traded the moogle E-A-R-T-H.
Now for the Regional eggs that are furniture, I'm unsure on the names of all of them, I just got the Wing Egg (San d'Oria) You have to trade the moogle the first 5 letters of a Region. If the depending on who controlls the Region, depends on which egg furniture u'll recieve. IE. I did Li'Telor, It was controlled by San d'Oria at the time, so I traded the moogle, L-I-T-E-L.
Some easy tips.
When U talk to the moogle, U can recieve a random letter'd egg each Vana'diel day. You can Also trade the moogle once each day to recieve and additional egg. You trade him either a soft boiled egg, hard boiled egg, color'd egg, or party egg. Doesn't matter which kind, and you can only trade one kind per day.
Now, If you have specific eggs you need. Its hell of a lot easier to get them by inviting party members to ur party who's name begins with the letter you need. When u speak to the moogle he will give u the option to take an egg that starts with the letter of the people in ur party or u can say no, and he will give u another random letter'd egg.
IE. If you had Bonkerskitty, Stormshadow, Jitsurou and Bunniebuffer in your party, the moogle will give u the option to pick and egg that that is either letter B, S or J. It will not give you the option to pick the egg that is ur first letter in your name.
ALWAYS THE MOOGLE WILL ONLY GIVE U ONE EGG PER DAY FOR TALKIN TO HIM, AND ONE EGG PER DAY FOR TRADING HIM A PARTY EGG, COLORED EGG, SOFT-BOILED EGG, OR HARD-BOILED EGG.