I found the game I was thinking of. I had to go through the
NES - Action on GameFAQs, looking at games I didn't recognize by name one by one, but I found it.
My memory was fucked. You don't fight a spring-type enemy; You beat most of the bosses by stomping on a spring-type thing instead of killing them directly. The second boss, who has an elongated neck, is one of the exceptions.
This game is weird as hell and the game physics are kinda fucked, but it's pretty fun.