The definition of the word exigency is an urgent demand or need for something. For example if you had an exigency for potatoes, you would need potatoes.
emergency
The coming hurricane is an exigency for residents in this island.
Depending on the context, alternate or exigency are possibles.
She had to leave her job due to military exigency
The state of being exigent; urgent or exacting want; pressing necessity or distress; need; a case demanding immediate action, supply, or remedy; as, an unforeseen exigency.
Both appear to be incorrect usages. The term exigency when referenced with servicemember FMLA relates to a personal crisis / hardship (emergency or pressing situation) imposed on the military member as a result of their service. The service isn't experiencing a crisis, it's the service member and their family who may qualify for FMLA under a "qualifying exigency"
Exigent is an adjective meaning exacting, demanding, requiring immediacy, urgent, or requiring more than is reasonable. Exigent can also be a noun, but that use is obsolete. "In exigent circumstances, law enforcement officers can enter without a warrant." "She resisted his exigent instruction that she forego dinner and stay late to finish the work so that he could leave early to go golfing."
What does exigency of service in the military mean/ does it mean to like quit? It means urgent need, something needing immediate action like a sudden call to active duty
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text, reader, author, constraints, and exigency