Now, forgive me as I know very little about the Greek language, but why is it that countries in the western roman empire (portugal, france, spain, italy, etc) all developed latin-esque romance languages, yet greek (as far as I know) stayed as it was with its own language and alphabet?

I think you mean why wasn't Greek replaced by Latin/a Romance language, right? This is a bit out of my time period, I'm a Byzantinist so hopefully I'll be corrected if I'm wrong. Latin actually didn't supplant languages other than Greek, such as Syrian languages (Syria, the Levant) and Coptic in Egypt. And both areas were sprinkled with major Greek metropolises such as Antioch in Syria and Alexandria in Egypt, the legacy of Alexander and Hellenism. And in the west, Romance languages weren't quite created by adding Latin to X indigenous language.

Compared to the west, the eastern Mediterranean was already densely populated by the time that the Romans arrived. And when they did come, conquests typically weren't in the manner of Gaul or Carthage. Some land was ceded peacefully (Pergamon, Cyrenaica), others were allies of Rome against some other local power that gradually were subsumed into the Roman state. Egypt with it's massive population came into the empire as Augustus' personal possession. Judea being an exception to the general mildness of Roman growth in the east.

Plus Greek was a prestige language, as it still is today to some degree along with Latin. It was the language of the classical philosophers and of Alexander, the upper class learned it. Later it was the language of the New Testament.

I ask this question since greece was under the rule of the byzantine/roman empire for almost 1000 years longer than any of the countries of the western roman empire, so I would have though it would have developed a romance language too.

That's literally true but deceptive because Latin rapidly fell out of use in the Byzantine era (see below).

I do know that at one point the byzantine emperors stopped speaking latin and started speaking only greek, but why did this occur?

Well by then (the 7th century), there wasn't much left of the Empire that didn't speak Greek. Egypt and Syria were lopped off by the Arabs, most of the Balkans were lost the Avars and Slavs. What was left was the major cities of the Greek mainland, the islands, and Anatolia. There was some land in Italy, but much of that (Sicily, the heel and toe of the boot) actually had spoken Greek for many centuries due to Classical Greek colonists. And soon the Lombards and the Pope would take the remaining lands in northern Italy. So it just didn't make sense to use Latin any further.

Some Latin did survive in official or technical language. Some terms were transliterated directly like "praitor" for praetor, others were translated like "sebastos" for "augustus."

A similar thing happened after the Norman Conquest of England. The invaders spoke French, but the Kings and Queens of the UK now speak English like the rest of the population does.

And why didn't something similar happen in the western roman empire?

Do you mean why didn't Greek move west under Rome? Greeks weren't immigrating westward, nor did the government push Greek. So why would it?