When I got my lower back injury and struggled with squats in general, I adopted the incline leg press as my primary quads exercise. Though, to be fair, I've always struggled with the squat because of my anatomy: my torso is very long relative to my femur so I have to compensate a lot and that's not good, so I decided to do them less frequently. I actually found it to be safer (incline leg presses) and actually managed to gain way more size by steadily increasing the weights week by week.
Hack squats are awesome alternatives that are also safer and easier to execute with correct form. It also targets the rectus femoris better so it builds some nice size higher up the thigh.
If you're determined to still squat, just without using your back, you can try front squats (as a previous poster already mentioned). You can also do squats while holding heavy DBs or KBs by leaning against a Stability Ball against a wall. Also, the closer your feet are together while executing the deadlift, the more the quads are targeted rather than the glutes and hammies, so close-stance deadlifts or using the duck stance (pointing your feet outwards at 45 degrees with only your heels touching) may be an alternative worth a shot just to get some extra quad work in.
Finally, there's single leg squats and single leg exercises that are great bodyweight exercises for increasing stability, control, and strength. Muscle and Fitness has a few cool articles on this:
At the end of the day, there is really plenty you can do without using a barbell over your back for legs, so I wouldn't stress