Frankly I think your situation could use a little more thought. A makeshift squat station, provided you are able to drop the bar on the floor if necessary, and especially if you’re doing front squats, would be a cinch to rig up. If you’re anything like me in terms of height and flexibility, you could just set up some saw-horses or benches at belly button height and start from a bottom position on the first rep.
Furthermore, there are exercises you can do to overload the necessary muscles that are equally difficult but require less weight: lunges (especially with the bar in the front squat position, or with DBs), overhead squats (if you aren’t full of shit about having huge quads, these might be just what you want: tons of strength, full body benefits, and they require all of 120lbs for most people), and very close stance deadlifts (which hit the quads pretty decently) come to mind.
But to go along with your question for the heck of it, and because improving the clean is a noble goal, here are a few things that help the clean IMO.
One-arm DB clean (be sure to take it from the ground and use a neutral grip) - this exercise could even benefit you if you go light and throw them in at the end of the workout
High pulls - be sure to use a proper triple extension, and don’t “meet the weight” - pull up and back at the end
Snatch (or clean) grip pulls - similar to a deadlift, but with more pronounced extension of the hips after the knees and a shrug and calf raise at the end, see a Staley article entitled pull your chain (actually a good resource for you, period)
Light “form focused” FULL cleans…I don’t practice these because I prefer the benefits of power cleans, but your full clean should be much higher than your power clean if you practice it. Most people can’t full clean their way out of a paper bag. Use 50 percent of your max clean or less for these
If you had the equipment, Thibs had an article a while back on functional isometrics that was good, but that would be taking it a little far if you weren’t a competing o lifter.
Dan John’s ebook From the Ground Up