A simple solution is to put some hefty eye bolts into a joist in the ceiling or a beam and hang the barbell by two ropes just a bit lower than you squat. Hey, we don't want to become deformed freaks from doing squats. Make sure you roll a towel around the bar for padding and do more reps with less weight to avoid this. Lunges are like squats in telemark position. They focus on one leg at a time and therefore you can use less weight on your shoulders and neck.
Balance is a little trickier.
You can avoid the barbell on the shoulders by doing squats or lunges with dumb bells hanging straight down in your hands. Here you need to invest in some heavy enough dumb bells if you are working out at home though, and it would be hard to increase the weight as you get stronger unless you owned a whole set of heavy dumb bells. A leg extension machine is useful in place of squats and lunges. The nice thing about such a unit is that it really minimizes stress on your knees. Many people with knee problems do leg extensions to develop their quads.