I won't bet my hand for this but seems to be ok:
ClearAll[return];
return[x : {0 ..., 1}, list_] := {#, list[[;; #]]} &[Length@x];
return[x_, y_] := {Length@y, y};
sqPeriod[list_] := return[FindLinearRecurrence[list], list]
sqPeriod /@ {
{19, 6, 19, 6, 19, 6, 19, 6, 19, 6, 19, 6},
{73, 7, 4, 73, 7, 4, 73, 7, 4, 73, 7, 4, 73, 7},
{73, 7, 4, 7, 2, 6, 7, 2, 7, 73, 9, 17, 7, 7}
} // Column