Does Layla have some sort of connection to the X777 event?
I don't think so, if you take a look at the Fairy Tail timeline you see that for some events there is no exact date known.

According to the timeline, these events happened somewhere in the year X777:


Romeo Conbolt is born.
Lector is born.
Layla dies of an unknown illness.


Therefore we can assume that the aforementioned events did not happen on July 7.

Underneath that you see the events of July 7 X777,


All dragons disappeared on 7 July.
Natsu is found by Makarov and invited to join Fairy Tail.
Mystogan left Edolas through an Anima portal and enters Earth Land.
Natsu Dragneel, Gajeel Redfox, and Wendy Marvell's Dragon parents leave them.


I think there is a bigger chance it had to do something with Mystogan coming to Earthland.