You were nearly there:
Jw1e1 = {{16, 832}, {32, 3150}, {64, 12237}, {128, 48212}}; J1 = ListLogPlot[Jw1e1, PlotMarkers -> (Style[Framed[H, RoundingRadius -> 20], Red]), Frame -> True] 
The nice thing here is that you can play around with RoundingRadius and that longer/larger text markers will be properly framed as well.
J1 = ListLogPlot[Jw1e1, PlotMarkers -> (Style[ Framed["Longer than strictly necessary", RoundingRadius -> 20], Red]), Frame -> True] 
Driving it home:
J1 = ListLogPlot[Jw1e1, PlotMarkers -> Framed[Style["Larger than \n strictly necessary", Large, Red], RoundingRadius -> 20], Frame -> True] 