I'll define a boreal forest as a place that
- Has trees, i.e., long-lived woody plants that are capable of growing at least ten meters tall and that grow both upward by extending new branches and outward by widening of the trunk. Amongst other things, this rules out times before ~380 million years ago, which was when the first trees formed.
- Has sufficient trees so as to constitute a forest, which I'll define as a largish area where trees grow sufficiently dense so as to form a more or less closed canopy. This distinguishes forests from areas with only a few trees such as savannas and krummholz.
- Has very harsh winters, with at least one month where the average temperature is well below freezing, and temperatures of -40° C are not rare. This distinguishes boreal forests from cold oceanic forests such as the Magellanic subpolar forests in southern Chile and Argentina.
- Has mild summers, with only a few months where the average temperature exceeds 10° C. This distinguishes boreal forests from hemiboreal and temperate forests. Note that some scientists do not make this distinction, classifying Köppen climate zone Dfb as boreal.
- Is extensive. This distinguishes large boreal forests from high altitude subalpine forests that would locally pass the above tests. Subalpine forests can occur at any latitude, including Australia's Snow Mountains, New Zealand's Southern Alps, and parts of the Andes. This is not a clear-cut boundary. As a climate cools, subalpine forests may spread to the valleys between mountains and then spread out beyond the mountains. At some point, such montane forests becomes boreal forests.
The need for very harsh winters coupled with the need for extensiveness means that boreal forests did not exist between the start of the Triassic ~252 million years ago to the end of the Eocene ~34 million years ago. The Earth was in its greenhouse phase during that long stretch of time, with balmy temperatures year round even at the poles. A boreal forest can only exist when the Earth is in its icehouse phase. This includes the last 34 million years, and also the period between 340 to 250, the late Paleozoic glaciation. Unlike the current ice age, the Earth in the late Paleozoic had land (Gondwana) surrounding the South Pole.
The question then is whether either of those intervals (the last ~34 million years, and the late Paleozoic glaciation) have supported a southern boreal forest. Per Rees-Owens et al., the answer is a possible yes to the more recent interval. Forests existed on Antarctica until as recently as 2.5 million years ago. Whether these were Köppen climate zone Cfc (subpolar) or Dfc (continental boreal) is very hard to say. If one relaxes the requirement for cold winters a bit, then those recent forests almost certainly were boreal.
Per Horton, et al., and also per the authors of several related papers, the answer is an unequivocal yes to the more ancient interval. Moreover, the boreal forest of that ancient time would have been very much like the boreal forest of present time: One of the largest ecozones on the planet.
Horton, D.E., Poulsen, C.J. and Pollard, D., 2010. Influence of high-latitude vegetation feedbacks on late Palaeozoic glacial cycles. Nature Geoscience, 3(8), p.572.
Pross, J., Contreras, L., Bijl, P.K., Greenwood, D.R., Bohaty, S.M., Schouten, S., Bendle, J.A., Röhl, U., Tauxe, L., Raine, J.I. and Huck, C.E., 2012. Persistent near-tropical warmth on the Antarctic continent during the early Eocene epoch. Nature, 488(7409), p.73.
Rees-Owen, R.L., Gill, F.L., Newton, R.J., Ivanović, R.F., Francis, J.E., Riding, J.B., Vane, C.H. and dos Santos, R.A.L., 2018. The last forests on Antarctica: Reconstructing flora and temperature from the Neogene Sirius Group, Transantarctic Mountains. Organic Geochemistry, 118, pp.4-14.