It's a joke, a pun, and it is a cultural reference.

There was a big hit song of 1963, sung by a singer named Lesley Gore, called "It's My Party and I'll Cry If I Want To." The singer is crying at her own party because her boyfriend left with another young woman:

https://www.youtube.com/watch?v=mCPqaG8sVDE

Nobody knows where my Johnny has gone,

But Judy left the same time;

Why was he holding her hand

When he's supposed to be mine?

It's my party, and I'll cry if I want to,

Cry if I want to, cry if I want to--

You would cry too if it happened to you!

The song is still fairly well-remembered and the title is a catchphrase. 

The word "whine" has the same vowel sound as "cry" so it sort of fits. One can imagine contriving a joke with a punchline like "It's my party and I'll die if I want to," or "It's my party and I'll fry if I want to." 

I forgot to mention: to "whine" means "to complain a lot, for a long time, in an annoying way, about something that isn't very important."

Imagine a six-year-old child walking with his parents and saying, over and over and over again, "When can we have ice cream? I want some ice cream. Let's stop for ice cream. Buy me some ice cream," and the mother says "Stop whining or I'll smack you!"

I think the British call it "whinging."

"Whine" is pronounced the same as "wine."

In this instance it means to complain about the same thing repeatedly. Sometimes, children will whine (repeatedly ask for) something that they want if they don't get their request granted.

Whine also refers to the high-pitched sound which a dog may make if it is unhappy.