So I planted them in two locations to see if they would survive. They all did. In both locations. Without any special care or winter protection.
Their range in the US is limited to zones 6 to 9 but like Impatiens and other Begonias we plant outside for the summer, they can be dug and divided to be protected over the winter and put back outside for the summer.
From Asia originally, they prefer moist spots, spread across the ground with runners and tiny plantlets. Those delicate red runners and tiny plantlets really show off to advantage in a hanging container. Overwatering and too much sun will make Strawberry Begonias suffer so be sure the spot or pot is well-drained and out of full sun.
Of all it's common names, my favorite is Roving Sailor! There are hybrids/cultivars if you want to stray from the species: Maroon Beauty, Tricolor.
Plants: Forest Farm has Maroon Beauty. Glasshouse Works has Tricolor.
Some Japanese culinary dishes have Strawberry Begonia leaves and flowers in them.
Plant Delights says Saxifraga is Latin for stone breaker, indicating the stolons ability to grow between rocks and their roots cracking the rocks open as they grow. And that rock garden reference is a reminder about their need for drainage!