According to Rav Avigdor Miller, they both came at the same time.

Putting that aside, the news is reporting that a chicken shortage is expected. Hundreds of thousands of chickens in coops in southern Israel, have died over the past week. It isnt clear to me if they were direct victims of terror (such as coops being burned to the ground or shot up with bullets and grenades), even if not the primary targets, or if it was an unintended result of people not being able to care for them over the course of the past week. Regardless, this is expected to cause a serious short term shortage which will cause prices to rise as much as 30%.

Truth is, the shortage is already felt in the stores. People were saying the dearth of fresh chickens in the stores was because of the shortage of Arab truck drivers and of Arabs working in the slaughterhouses (among other places), but perhaps it has also been affected by this.

I did not see anything about an upcoming egg shortage for the same reason, but I would assume if there is a shortage of chickens there will also be a shortage of eggs soon after.

But theoretically one could ask, which came first - the chicken [shortage] or the egg [shortge]?