Leatha,

The Republican Blues were Co. F of the 11th SCVI, part of Hagood's Brigade. Unit badges and insignia were practically non existant in the Civil War. The only units which had a regular badge were pre-war militia units which went into active service when the war began. The Confederate Army did not have a system of unit badges. If the Republican Blues had a badge it was one they adopted for themselves. I think it highly unlikely they had an insignia. However there is a book available about the unit it is:

The Blues in Gray: The Civil War Journal of William Daniel Dixon and the Republican Blues Daybook, by william Daniel dixon and published by the University of Tennessee Press.

Len