Grabbed this today on my elemental shaman, solo. Took a few tries to work out the mechanics but was straightforward enough I'd expect any class could do it if they have some LFR gear. What I found worked best was to start with the eggs furthest away from the orb and work back towards the orb on the second channel. This causes more of the mobs to attack you. Also, use the dragonkin sleep ability when they come out.
1843 pets
Grabbed this today on my elemental shaman, solo. Took a few tries to work out the mechanics but was straightforward enough I'd expect any class could do it if they have some LFR gear. What I found worked best was to start with the eggs furthest away from the orb and work back towards the orb on the second channel. This causes more of the mobs to attack you. Also, use the dragonkin sleep ability when they come out.