Best results for improving your bench and squat are to bench and squat heavy. It sounds, however, like you want to improve your barbell bench and squat strength by using bodyweight methods. This will be difficult. Here's what I would try:
- Get good at pistols. As soon as you can do five in a row, add weight. Anything you can hold in your hands will work. Heavier is better.
- Push-ups. Increasing the number of reps beyond, say, thirty or fifty or a hundred isn't going to much help your bench press, so look into harder gymnastic variations of push-ups once you're at that point.
- Do you have a workout partner? Put them on your back and squat.
- Handstands and handstand push-ups. The carryover to bench is not significant, but increasing your pressing strength might help somewhat.
If you're not looking for squatting or benching numbers specifically, but are interested in strength using bodyweight methods more generally, I recommend looking deeply into gymnastics. Coach Sommer's Building the Gymnastic Body is a good place to start, as is Ross Enamait's work.
But if it's really squatting and benching you want, best results will come from finding a barbell, weights, and rack for cheap on Craigslist or a yard sale.