イヤーンバカーン is an cliched, interjection-like set phrase. いやーん (嫌ーん) literally means "no" or "I dislike it" and ばかーん (馬鹿ーん) is "idiot." It's not a refusal but rather an aged, stereotyped word used when a woman toys or trifle with a man. So the person who said it is childishly and jokingly mimicking a sexy adult woman.
Probably it's a wordplay similar to a classic joke 当たりの前田のクラッカー. It's a type of pun-like wordplay made by combining a totally meaningless phrase. The more meaningless and irrelevant phrase you join, the funnier it will sound. It's technically called a 地口. So that いやーんばかーん part does not really carry an important meaning, and you may even ignore it.
アンハッピーニューイヤー is "Unhappy New Year" rather than "A Happy New Year." That should be another cheesy joke he made. That アン followed by ♥️ is like "oh" or "ah" induced by sexual stimuli.