First of all, there was no "Russian" side in the Finnish Civil War. The war was fought between White and Red Finns, and while the latter did get some help, volunteers and equipment from pro-revolutionary Russian troops that had been stationed in Finland prior to Finnish declaration of independence in December 1917, Russia as such did not participate in any official capacity and the demobilization of Russian troops made them fairly reluctant to participate in the "Finn's war." Nevertheless, White propaganda sought to portray the struggle as one against "russkies and their lackeys" - partly because that sweeped nicely the "crushing the peasant revolt" aspect of White campaign under the proverbial rug, partly because White leadership had grave doubts as to what extent conscripted smallholders from Northern Finland would be motivated to fight Southern smallholders of equivalent economic and social class. The idea that the Civil War included significant Russian involvement comes from that propaganda. [1]

That said, it is in my opinion likely that during that time even knowledge of Kalevala as a national epic was limited to Finns. Kalevala is indeed based on old Karelian oral tradition, but it was significantly edited by Elias Lönnrot, who first thought to collect those disparate poems and publish them in 1835. Kalevala is acknowledged to have been a very important linchpin in the Finnish nationalistic movement, "a cornerstone on which national identity was built" [2].

However, as far as I'm aware, similar nationalistic movement did not really exist (or get off the ground) in Russian Karelia. From what I've read, Karelia was not a nation anywhere close in the same sense as Finland was in 1917. Karelians were of course aware of their own oral tradition, but it was tradition (and fairly fragmentary at that) and not a national epic. There was an insurrection by "forest guerillas" in 1921 against the "people's commune of Karelian proletariat," but the rebellion failed, despite help from Finnish volunteers [3]. That's the extent of "true" Karelian independence movement I'm aware of, although I'm the first to admit this is not really my area of speciality.

To Finnish volunteers participating in the Aunus expedition and other adventures, Kalevala was an important part of the nationalistic lore they had been inculcated with since they were schoolboys. In the mostly right wing circles where the expeditions were hatched, Kalevala was one of the main ideological justifications for "Greater Finland" that would have included Eastern Karelia -- the mythological "song lands" and cradles of Finnishness. The same goes perhaps even more forcefully for soldiers during the Winter War: nationalistic ethos was strongly present in primary and secondary schools and in higher education, although the extent to which it stuck to individuals of course varied. But the loyalty of these people was strongly to Finland: the more extreme just thought that Finland should include Eastern Karelia as well. (Although it should be noted that in the 1920s, the idea of "Greater Finland" enjoyed considerable popularity in more moderate circles as well, with only Social Democrats willing to forgo Eastern Karelia altogether. The difference was that moderates thought the border should be determined by Karelian plebiscite rather than force. [3])

I believe it is very likely there were also Karelian nationalists but I do not have any good sources to that at hand. What I recall from more general history is that Karelian activists, including people simply interested in Karelian folklore, fared badly in Stalin's Soviet Union and were basically destroyed wholesale during the Purges. The Soviets attempted to form a puppet Terijoki government in the opening stages of the Winter War and on paper formed the Finnish Army Corps; but most of the potential volunteers/conscripts had been killed off in the 1930s, and there are reports that random people from Soviet Karelia and even Leningrad were simply press-ganged to serve in the "Finnish" army [4]. It is unlikely these were very well versed with Kalevala as such hobbies were certain to be suspicious in the 1930s.

Sources

in Zetterberg (ed.) (1987) Suomen historian pikkujättiläinen, WSOY, Helsinki:

[1] Salokangas, p. 614. [2] Zetterberg & Pulma, p. 460. [3] Salokangas, p. 632.

[4] e.g Stepakov & Orehov (1992) "Paraatimarssi Suomeen:" Talvisota venäläisin silmin. WSOY, Helsinki.