I figured out how to make this work. I was able to embed icons instead of the text. Here is how it looks now, which is exactly what I was trying to do.
Here is the code:
<calcite-segmented-control id="cboStylePoint" selection-mode="single" scale="s" width="full">
<calcite-segmented-control-item value="circle" checked><calcite-icon icon="circle" scale="s"></calcite-icon></calcite-segmented-control-item>
<calcite-segmented-control-item value="plus"><calcite-icon icon="plus" scale="s"></calcite-icon></calcite-segmented-control-item>
<calcite-segmented-control-item value="diamond"><calcite-icon icon="diamond" scale="s"></calcite-icon></calcite-segmented-control-item>
<calcite-segmented-control-item value="square"><calcite-icon icon="square" scale="s"></calcite-icon></calcite-segmented-control-item>
<calcite-segmented-control-item value="triangle"><calcite-icon icon="triangle" scale="s"></calcite-icon></calcite-segmented-control-item>
<calcite-segmented-control-item value="x"><calcite-icon icon="x" scale="s"></calcite-icon></calcite-segmented-control-item>
</calcite-segmented-control>
Cheers!
Tom