I'm not from Macedonia, although I know people there. Both of your sides need to read up on your history. Macedon was located between the traditional Greek city-states and the Thracian tribes (and others like the Illyrians), and had a culture and language closer to that of the Greeks but with strong Thracian and Illyrian influence (for example, they used both Greek, Thracian, and Illyrian names). The ancient Macedonians sought recognition as "Greek" from their southern neighbors, as Greece was the heart of wealth and culture; by contrast, the ancient Greeks debated heavily among themselves whether Macedonians were actually Greeks or not and many were not willing to accept them. The issue was only settled they were conquered by Philip (obviously not wanting to say that they had been conquered by barbarians ;) ) The Macedonian leaders' habits of adopting the cultures of the countries that they conquered made it a relatively moot point anyway. Macedon was near present day Thessalonica. The country of Macedonia's claim to the history of Philip and Alexander is pretty weak; they did not extend their empire particularly far up the Vardar / Axios (at the time, Illyria), and where they did they stayed near the river. However, future rulers of Macedonia did. By the time of the Roman conquest, what was Macedonia had become modern Macedonia plus modern Albania and the northern half of Greece. This become the Roman province of Macedonia, which existed for hundreds of years. Classical Greece remained its own separate entity under Roman control, Achaia.
Let's repeat: Modern Macedonia was the center of the Roman province of Macedonia for hundreds of years. Yes, they have a right to the name.
During the Byzantine times, a series of waves of Slavic invaders (the most powerful being a later wave, the Bulgars) moved into the area, overrunning not just today's Macedonia, but the entire interior of Greece. Their control of these areas lasted hundreds of years and they interbred with the local populations. Greek speakers retained control of the coasts, and eventually re-expanded back into the interior; the populations there were subsequently re-Helenized.
The area that is modern day Macedonia was subsequently traded off between one empire and the next all the way up to the modern period. Greece, after gaining its independence from the Ottomans (again, initially only in the southern portion of of what is modern Greece - the part that was traditional, pre-Macedonian-era Greece), progressively took over Ottoman lands to the north and northeast in the 1800s, expanding into the area formerly occupied by the city-state of Macedon, and even the areas once occupied by the Thracians. These areas were steadily Helenized, especially in the 1920s with the influx of large numbers of Anatolian Greek refugees - 270 thousand in Thessaloniki alone.
The basic summary of it is: there are no ancient Macedonians anymore, and nobody has a "pure" claim on the name. But both sides have ancient Macedonian "breeding". Neither speak the ancient Macedonian language (even the ancient Macedonians stopped speaking their language by the 3th century BC), although Greek is certainly closer. Both Greek Macedonia and the Republic of Macedonia are located in the heart of areas called Macedonia for centuries - Greek Macedonia being the heart of the original Macedonian empire, and the Republic of Macedonia being the heart of the later kingdom and the Roman province of Macedonia. So yes, you have every right to criticize the Republic of Macedonia's cooption of Alexander and Philip. But you're in the wrong when you try to act like they have no claim to the name.