Sure, just about any exercise can be done one side at a time for muscle/strength improvement (even a DB bench press). Gives more focus to any targeted muscle group. That video showed the floor version of the one legged squat, which a lot of gymnast do in training. This version also gives more flex/stretch to the ankle & hip area, which an athlete needs. Though some people have a hard time with it no matter how long & hard they try. Could put the heels on a 2X4, BB plate, etc for better balance, as some do. If doing that, you will not get that extra stretch to the ankles/hips as doing it flat footed.
Another version is doing the one legged squat off a bench, the higher the better. May find it easier to balance yourself. More so when you begin to add weight (BB or DB) to the exercise.
Though not a one legged squat (in the true sense of the word), there is the Step-up. Which can offer more involvement from the hams/glutes because you are stepping forward and up rather than pushing straight up as with the squat. Might be a much better choice, for a lot of folks, over lunges if more work for the hams/glutes and hips is required. Many athletes are including Step-ups into their training. And using quite a large amount of weight (either BB or DB) in their workouts.
The regular squat is a wonderful exercise but a one legged version or Step-Ups should also be included in any workout plan. Can give a better muscle balance to leg development plus an equal amount of strength to both legs. Good Luck.