1. What does it mean?
Short for 얼어 죽어도 아이스 아메리카노: roughly, 'iced Americano even if I freeze.' It is an exaggerated declaration of loyalty to an iced coffee order, including on cold days.
2. The nuance
The dramatic wording is a joke about preference, not a literal statement about danger. It refers particularly to iced Americano, although people sometimes extend the joke to their preference for cold coffee more generally.
3. Examples
“눈이 와도 내 주문은 아이스 아메리카노야. 얼죽아거든.”
Even when it snows, I order an iced Americano. I'm an iced-coffee-in-any-weather person.
“오늘은 너무 추워서 얼죽아도 잠깐 쉬고 따뜻한 거 마실래.”
It's so cold today that I'm taking a break from my iced-coffee loyalty and getting something hot.
Sources
Korea Policy Briefing: K-Gonggam
References support the meaning and usage. Example sentences are written for NoCapWiki.