BSS hits much more the glutes than the quads. But a good squat uses the glutes a lot. And either way, yes, there will sure be some carryover to squat, just don't expect a 2*BW ( 130kg ) when you were at 60kg before.
For progressing BSS more:
1) Add volume. You did 40kg for 1-1-2 reps, you have to get to 3x8-10
2) Volume BSS with heavy DBs gets hard. If you do 8-10 reps each leg the set lasts too long, there are grip problems. What i suggest ( and have done it myself with great success ) is using the smith machine. It eliminates the balance and setting up problems and you can focus on your lift. Even adarq approved it back when i was doing it, he said 'if the smith machine is worthy for 1 exercise, this one is BSS'
3) Make sure you are not sacrificing ROM and form for weight. Keep torso as upright as possible ( you can half-BSS / half-RDL it, which is wrong ) and go down until knee almost touches the ground. This is another aspect that the smith machine helps , you can't cheat bending forward.
My gym is too small, it doesn't have a smith machine unfortunately, it has 6-7 cardio machines, and 2 squat racks and 1 bench press. that's all, but I try to keep upright only when i reach those struggling reps, I lean forward a little when going up.
But it does take long, to do 3 sets of 8-10 reps for each leg, I can only manage 2 sets and move onto next workout to fit it in before gym closes.
But today's session was good did 2 x (4,4), (4,3) => (L, R) first few reps good form last few reps hard and the form goes bad, but I am going to stick with this weight until I can do 8-10 reps, I probably will be able to fit in 2 sets, then will get less time to do single leg RDL's they are hard because after BSS my quads are sore that when doing the single leg rdl my quad's soreness is making it difficult to do the workout mainly for my right leg, left leg is fine.
weird times I got today for my 60m sprints alternating lead leg.
1. 8.12 L
2. 8.13 R
3. 8.12 L
4. 8.12 R
with the BSS it was tough for my right leg, I was sweaty I would manage a half and then my other leg comes off the bench, so I try again, I fail dropping it, try again, half, soo much sweat trickling down my face, I just have to complete the reps for my right leg, until I know my legs are finished and cannot take anymore so I move onto single leg RDL, the same balancing issues.
Rating: 7/10