I'm not a doctor, but pressure buildup and capillary burst doesn't sound healthy. Maybe you're doing too many push-ups in one set? If not, you could try some progressive inversions and then move on to the handstand push-ups.That would help your body get accustomed to the pressure and not cause damage.
My suggestion would be to hold downward dog and wide-standing forward bend for 30s and 15s each, and once you're comfortable, move into a handstand for 10s. Gradually, over some time, increase the duration. Once you are ok in the handstand for about 20-25s, then try one pushup. Over time you could increase the number, but giving your body time is the key. Don't do too much too soon. I don't know if you're following some training on handstand push-ups, but it's very important to follow a forward bend with a backward bend, a great one being the camel pose.
If you're doing this exercise with a trainer, follow them. If not, you should know that full-body inversions can cause high blood pressure and aggression if done too long/without doing the counter-pose. Very athletic people may get away with it, but for the common man the risk is real.
I can do shoulder stands and head stands for about 5 minutes, but never do more than 3 handstand push-ups in a set. This is not a standard by any means, just indicative of how much a normal body can take.