The following graphs are given in the graph6 format. The graphs were found using geng and SAGE.
All squco graphs of order at most 11:
Order 1-6 (The singleton graph)
Order 7 (The 7-cycle)
Order 8 (The graph $C_7[2,1,1,1,1,1,1]$)
Order 9 (The graph $C_7[3,1,1,1,1,1,1]$)
Order 10 (The graphs $C_7[4,1,1,1,1,1,1]$ and $C_7[1,2,1,2,2,1,1]$)
Order 11 (The graphs $C_7[5,1,1,1,1,1,1]$ and $C_7[2,1,1,2,1,2,2]$)

All triangle-free squco graphs of order at most 12:
All squco graphs of order at most 11 are triangle-free.
Order 12 (The graphs $C_7[6,1,1,1,1,1,1]$, $C_7[2,1,1,3,2,2,1]$, and the Franklin graph)

All bipartite squco graphs of order at most 15:
Order 1-11 (The singleton graph)
Order 12 (The Franklin graph)
Order 13 (3 graphs)
Order 14 (11 graphs)
Order 15 (74 graphs)