Place graph vertex labels in circles

You can change Rectangle[...] to Disk[{xc, yc}, Min[w, h]] or use any other function of w and h to control the disk radius.

But ... you can also get the same result without having to define a custom VertexShapeFunction using the options:

options = Sequence[VertexStyle -> LightYellow,
   VertexSize -> 0.15,
   VertexLabels -> Placed["Name", Center], 
   VertexLabelStyle -> Directive[20, Red, Bold, Italic], 
   GraphLayout -> "CircularEmbedding", ImageSize -> 350, 
   EdgeStyle -> Blue];

Graph[{1 <-> 2, 2 <-> 3, 3 <-> 1}, options]

You can also use the option PlotTheme -> "IndexLabeled" in versions 12.0+:

Graph[{1 <-> 2, 2 <-> 3, 3 <-> 1}, PlotTheme -> "IndexLabeled", 
 VertexStyle -> Yellow, VertexSize -> Medium, 
 VertexLabelStyle -> Directive[Italic, 16], EdgeStyle -> Blue]