Are bodyweight squats, push-ups and pull-ups a decent substitute for the starting strength routine by Mark Rippetoe?
Not really, because one of the most important elements of Starting Strength is progressive overload by increasing weight. With barbells, it's easy to add small amounts of weight every workout, causing your body to adapt and get constantly stronger.
However, with body weight exercises, you can't really add weight. Instead, you must add reps or use harder variations to provide progression. Compared to adding weight to a barbell, these options are less precise and more unreliable. This makes any body weight routine materially different from Starting Strength.
You can potentially get a good workout with a body weight routine, but it's not going to look like Starting Strength.
Let's look at the three exercises in more detail:
- Squats will be particularly problematic. Without adding weight, squats are far too easy. You'll be able to do so many reps that it's an endurance exercise, not a strength exercise. Perhaps you could get a better workout with single-leg squat variants or jumping exercises. But I have a hard time imagining good progression even with these variants. Legs are simply too strong to effectively challenge without an external source of resistance.
- Push-ups can be a good exercise for beginners, but once you get stronger, they have the same problem as squats with being too easy. However, unlike squats, there are some good routes for progression. You can increase the difficulty of push-ups by elevating your legs and varying your hand positions, or even using one hand. I'm no expert in calisthenics, but apparently some athletes get good results with this.
- Pull-ups are difficult enough for most people that they make a good strength exercise. In fact, the latest Starting Strength routine on the web includes pull-ups (see Starting Strength Training Programs). Doing 3 sets to failure 2–3 times per week is a good way to build upper body pulling strength. However, once you get past 8–10 reps, you should consider adding weight to keep it strength-focused rather than endurance-focused.