Learn Icelandic: Hvílíkir flugeldar!

Because hvílíkur acts like a strong adjective and must agree with the noun it modifies. Since flugeldar is masculine, plural, and in the nominative case, the determiner becomes hvílíkir.

Would it be correct to add the definite article and say Hvílíkir flugeldarnir!?

No, hvílíkur is always followed by an indefinite noun. Just like in English where we exclaim 'What fireworks!' and not 'What the fireworks!', Icelandic uses the base, indefinite form of the noun here.

Even though it starts with hv-, this isn't a question. How can I tell?

While many hv- words are interrogative (like hvað or hver), hvílíkur is an exclamative determiner used to show astonishment or emphasize a high degree. It translates to 'What a...!' or 'Such...!' rather than asking for information.

What is the literal breakdown of the compound word flugeldar?

It is made up of flug (flight) and eldar (fires, plural of eldur), so it literally translates to 'flight-fires'.