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 longer text markers will be framed as well.

J1 = ListLogPlot[Jw1e1, PlotMarkers -> (Style[ Framed["Longer than strictly necessary", RoundingRadius -> 20], Red]), Frame -> True] 

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]