I am doing my Japanese reading, and I came across this sentence:
快復の見込みのない末期状態になったとき、単なる延命治療．．．
I don't understand why we use の here instead of が. I am still a beginner so I always use 名詞＋がない. I asked my Japanese friend and he told me that "見込みがない is not bad but 見込みのない is more natural".
Another quick question is what's the difference between 快復 and 回復？
The reason why の can be used here is already explained in this question:
How does the の work in 「日本人の知らない日本語」?
Regarding the nuance, I think the difference between 見込みのない and 見込みがない in your example is very subtle, and both are perfectly natural. 見込みのない may only slightly sound more politely or euphemistically.
快復 is an uncommon alternative spelling of 回復, and 快復 can be used only when it refers to a cure from disease. 回復 is much more common and can be used for various kinds of recovery (from disease, economic depression, bad weather, etc.)
