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]

![](https://i.sstatic.net/oneyk.png)

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]

![](https://i.sstatic.net/Zeysu.png)

Driving it home:

	J1 = ListLogPlot[Jw1e1, 

	 PlotMarkers -> 

	 Framed[Style["Larger than \n strictly necessary", Large, Red], 

	 RoundingRadius -> 20], Frame -> True]

![Mathematica graphics](https://i.sstatic.net/RgckX.png)