Someone built the house you live in, makes the electricity to power it, and pipes water to it so you can drink and bathe. Someone made the car you drive and the road you drive it on. Someone grows the food you eat, harvests it, trucks it to you, and sometimes even cooks it for you and brings it to your house. If these people were not there, you would have to do all this yourself every day. You wouldn't have time to do anything else.
It's in your own personal best interest for all of these people to be healthy (so they can do these things), happy (so they will want to do these things), and educated (so they can do these things well). That's what universal health care and public education are for. Even if you, personally, never use these services, you're surrounded by people who would benefit from them - and when they benefit, so do you.
Some might argue that these people should pay for their own health care and educations with their earnings. That's certainly one way to do it - provided that they earn enough to afford the health care and education. Take the people who build roads. Everyone knows that the building and upkeep of roads is paid for out of taxes. Well, the workers are also paid for out of those taxes. If we want to give them a fair chance at remaining healthy, they must be paid a high enough amount to afford the health care - above the amount they need for food and shelter. We can either raise their pay, or give them the health care separately from the take-home pay. Either way, it's still the same taxes.
Or maybe we don't care if they have a fair chance at remaining healthy. We can pay them as little as possible because what they do isn't important, and they aren't important either, and they should be happy to work for free. It doesn't matter if they perish because we can just find more of them from somewhere else. So long as they stay out of our sight, we can happily live our own very important lives, secure in the knowledge that the world is running along fine. If it's not fine for someone else, well, that's not our problem, now is it?
Society is not "every man for himself." Everyone contributes something so that no one has to do everything. People naturally want to contribute. See to it that they have the means, take care of all the people in it, and society as a whole will prosper.