As far as I know myself, the Biegun/Begun name originated in Russia. As one story goes, when certain groups, among others, that became known as the Starovertsy or Old Believers, who denounced Church and State, ran or fled from government and religious persecution, they acquired the name of Beguny or Bieguny (which literally means runners in Russian): A particular place in Russia, the Cossak region, was a place that these groups fled to, and during the Zaporogian Cossak period, when, supposedly people did not go by formal surnames, the Cossak military (who were a government to themselves) would, for organizational purposes, name, or register people by names that depicted their identity in some way. Thus, the Biegun name stuck.
The Bieguny are described by Harold Whitmore Williams, in RUSSIA OF THE RUSSIANS, 1914: "The extremists amongst the Old Believers, the Bieguny or Stranniky were convinced anarchists, denied the State absolutely, refused to have any intercourse with the authorities, rejected passports, and were, in consequence, condemned to a life of wandering, of constant escape from the police; hence their name of Bieguny (runners). The Old Believers lived in an atmosphere of legend, dark superstition was very strong among them; they retained unmodifed old popular beliefs in evil spirits, and persecution added to their life a peculiar rigidity and gloom.... But they were men of conscience, lived very strictly, refrained from smoking, fasted often, and were extremely methodical in all their dealings. The consequence was that, like many other persecuted communities, they, as soon as the persecution became less severe, began to prosper exceedingly. They built up large business, and helped each other regularly as members of such close communities always do. A great many of the wealthiest merchants and manufacturers in Moscow now are Old Believers..... The Old Believers are a particularly interesting community because they preserve so many distinctive features of the Russian life of an older time. They have old ikons which are of great importance for the study of Russian art. Their mode of speech, their domestic habits, their superstitions serve as historical and ethnographical documents....... Who are all these wanderers, these men and women who bear strange names, the Pomoriane, Fedoseievtsy and Filipovtsy, the Bieguny, Stranniki, Molokane, Dukhobortsy, Khlysty, Skoptsy, Shtundisty, the New Israel and the non-prayers, mystics and rationalists, ritualists and protestants, wrestlers with the Spirit and mortifiers of the flesh? They deny each other fiercely, as fiercely as all of them deny the State Church, and each clings fast to the little lamp or to the smoking torch that for him lights a way through the darkness of this life. But the Bieguny, the Runners, are the prototoype of them all, those Bieguny who have no abiding city for they seek one to come. It is true that even these inveterate protestants against Church and State have now largely lost their energy of resistance, that only a few of them now live up to the full extent of their creed and take monastic vows and wander in the forests refusing to have any traffic with the representatives of a State that they consider to be a manifestation of Anti-Christ....The Bieguny have gone to the extreme of denial. They run ever that they may grasp the prize of their calling..."
From all this, it becomes apparent that the Bieguny were groups of people, obviously of different blood lines, genealogies, that eventually accepted the Biegun name that put them under a blanket of social identity; the name eventually stuck as a surname. Nevertheless, of course, a lot of the Bieguns are actually directly related.
I couldn't get into the site where you said you researched the Biegun name going back to 1575. As far as the Biegun name originating in Poland, all I can say is that the Cossaks fought in the 1600's against the Poles when Poland occupied part of the west Ukraine; thus you have a mixing of populations and cultures, as well as migrations.
I just remembered I found a book in the library written by a Biegun or Biegunov. This man happens to be an expert in old Russian language, I believe that is an etymologist. He may very well know (it's possible!) the origin of the Biegun name. If you're interested, I will try to find that book again and get his complete name. Maybe we can write him a letter!! Is this an example of Biegun fanaticism or what!