starting my day with pushups and squats

I suppose it depends on the intensity of the main workout you do in the evening but it would probably be ok. If your worried you could try and mix it up and do push up and squats every other day, and then something targeting another body part on the next day.

I heard of a method called "greasing the groove" recently, the idea behind it is you do the same exercise or exercises everyday, sometimes multiple times with the intention of increasing your reps/load over a short period of time. You avoid overtraining by never going to failure and tiring yourself out too much. It's meant to be good, check it out on Google if you get a chance.

I was thinking of doing something similiar daily to increase my pushup and handstand push-up because im shockingly bad at both!