First and foremost, welcome back.

Secondly, I'm confused as to what eggs you are talking about, as you didn't really provide any context. What is the name of the quest you are on? Did you also buy the story-skip potions in tandem with the job boost?

As for the gear... and this is pure speculation because I have no way to confirm this right now:

Since you bought the job boost potion, it is supposed to have completed all job quests up to level 60 for that class. This means you "did" the level 45 and 50 job quest for DRG. This means you should have unlocked the DRG artifact armor (the armor that is in the picture you provided). Odds are, you're probably going to need to talk to the Calamity Salvager to get it. The Calamity Salvager allows you to rebuy certain pieces of special equipment that you might have ditched for some reason. In this case, never technically got, but should have. Or, you might already have it in your armory chest. Though, I figure you'd know if you did by now.

Calamity Salvager's are at:
- Limsa Lominsa: Upper Decks - X11.3 Y14.3
- Old Gridania: X10.0 Y8.4
- Ul'dah - Steps of Thal X:12.6 Y:13.1

There may be two versions - the Draken set and the Wyrm set. Wyrm is slightly better than Draken. Both are WILDLY terrible compared to Shire gear. If you can get your hand on some glamor prisms, however, you should be able to wear the Shire gear but change their outward appearance to be your DRG AF1 armor. Provided you have the DRG AF armor in the first place. You can't glam something you don't have.

As for the weapon, that is the DRG relic weapon, and that's a whole different quest system for a weapon that is light-speed inferior to the shire weapon you got. However, if you want to get that one as well - and provided you bought the story-skip potion....

Relic Weapon Quest