Yes Essos will be threatened as well.
Long Night occurred some 8,000 years ago but it was not limited to Westeros only. Legends point to the same event all over the Planetos as there are similar legends found as far as Yi-Ti and Asshai. Melisandre knows about this event even though she is from Eastern most land of Asshai. If the others managed to threaten the Eastern realms back then, it is likely that they would be able to do so again.
According to GRRM in an interview:
“The Others can do things with ice that we can't imagine and make
substances of it."
In S06E10 we saw that Citadel sent White Ravens to all lords, which heralds the coming of winter. Winter means more ice and snow and thus increase in power of the Others.
Using their power of creating substances out of ice, they might be able to build an ice bridge across the Narrow sea (It is called Narrow for a reason) and wage war in East and West.
We do not know exactly how can they threaten Essos but we do know that they did so in the Past and thus have the power to do it again. Remember, Arm of Dorne, the sole land link between Essos and Westeros was broken some 4,000 years before the Long Night fell. Others managed to extend the war beyond the sea anyways back then and there is no reason why they should not be able to do it again.