if I do say only push ups for 2 days, does that satisfy above need? Say 20-30 push ups on each day with breaks.
If I am not mistaken, you found the advice here. The point of that page is simply to give the bare minimum amount of exercise that we should be doing. The more you do, the more benefits you will receive. On that page it mentions that you need to work out all major muscle groups. Pushups will primarily work out your pecs, shoulders, and triceps. This still leaves your legs, hips, back, abdomen, and the rest of your arms.
If no what is the minimum program for this section?
Again, I can't stress this enough, the more you do, the more benefits you will receive (up to a certain point, but it doesn't sound like you have the mindset to even get close to overtraining). Your question simply asks what is the minimum amount of work you can do to fulfil the governments recommendation. Although I believe that is the wrong mindset when it comes to fitness, here are a few which together can fulfil that recommendation:
Pushups - Chest, Shoulders, Pec, Triceps.
Body weight squats - Glutes, Quads, Hamstrings, Adductor, Hip Flexors, Calves.
Supermans, inverted rows, or pull ups - These are all back exercises that you should progress through depending on your fitness level. Supermans are easiest, inverted rows are a bit harder, and finally pull ups are the hardest.
Each exercise above slightly works the abdominals as well, so each muscle group that they recommend is covered.
To build strength you must increase the number of reps over time, this is called progressive overload. This means that simply doing 20-30 every day, forever, will lose the benefits that it first had. You will remain at the same strength unless you increase the repetitions/weight over time. Say you can do 30 reps max, then you could try doing 3 sets of 20 with 1-2 minute breaks between sets. The next workout you should try to increase one of the reps by at least 1. So instead of 20x20x20 shoot for 20x20x21. It is perfectly normal to not see improvements on many workouts, but the goal is to always be improving.
Although exercise is important, your diet is going to have a much larger impact on your health than any amount of exercise you can do. You need to have good food, proper amounts of sleep, and enough exercise to live a healthy life.