First of all, it's worth noting that Japanese has no 形容詞 or 形容動詞(な-adjective) which directly corresponds to the English adjective sick. (although you can say 「彼の具合【ぐあい】が悪【わる】い
」, if you don't mind replacing the subject)
We can say 「彼 は [病名] だ」、「[病名] の 人」、「 [病名] に なる」、where [病名] can be 癌 (cancer), 肺炎 (pneumonia), 糖尿病 (diabetes), 骨粗鬆症 (osteoporosis), or 病気 (disease) in general. And they are all nouns. We usually cannot say 「 [病名] な 人」, so 「癌な人」 or 「糖尿病な人」 is wrong.
I think that's why we often treat the loanword ホームシック also as a noun. Words such as lovesick or airsick have no corresponding adjectives either in Japanese; instead we have nouns 恋煩【こいわずら】い (love sickness) or 飛行機酔【ひこうきよ】い (air sickness).
That said, I think this rule is somewhat loosened if certain 外来語 diseases are concerned. In fact, both 「ホームシックな時」 and 「メタボな人」 sound as natural as 「ホームシックの時」 and 「メタボの人」 to me. Japanese tend to create many な-adjectives from English adjectives, as in スマートな, リスキーな, etc. They can even create な-adjectives from English nouns, such as ワンパターンな, ハイリスクな. I see no reason why homesick cannot act as a な-adjective as well as a noun.
Lastly, some people may argue that Japanese young people use "病気な人", or more commonly, "ビョーキな人". Such use of 病気 is a slang and has a special meaning (abnormal, crazy, mad). Use "病気の人" for someone with a real medical disorder.