My professor argued that since our parents did not ask for our permission to be born, all of us should be entitled to three things at birth: free housing, free healthcare, and free education (includes college). What are the pros and cons?

Sounds like your professor is just trying to increase demand for his or her own profession. College would be wasted on a baby. The other three things we already get. Babies in college? Ridiculous!

As long as the obligation rests with the parents, who did do something voluntary to bring you into existence, then I agree. In other words, it's free from your perspective but that doesn't"necessarily" accrue to society. Note that I personally think that society benefits by reducing suffering, but not because of the reason provided.

I assume he meant that your parents `should pay the cost of your housing, healthcare, and education since they were the ones who had a say in you being born (making exceptions for those fathers who did not have a say - such as rape or the mother impregnating herself with a used condom).  How long, then,should the parents be responsible for providing these things and why does that responsibility end?

Nope, rather the implication is the parents should provide for their children at least until they are adults. The parents did not ask the permission of the taxpayers either, so why should the taxpayers be on the hook to provide for them? Once the government stops subsidising those who cannot stop breeding like dogs, then automatically the government dependent population will decrease.

Many people, including me, believe not only do we decide to be born we also pick the circumstances of our lives. But even if this isn't true it must be a specious argument calculated to spark debate.  The idea is so ungrounded in reality that an educated person would have to have a screw loose to make it.

Maybe your professor is thinking on a different level and if he does then I agree with him. Here's what I mean: Remember, in the original world, it did not cost anything to live and it enabled early people to actually live, enjoy, and discover the world freely. Everything was free for all living things. Then man "the most intelligent" animal, decided to sell things that was given to him for free. Then it all went down hill from there. Now we don't live, we work. We are the only animal that pays to live. I was not told I had to pay to live here. If I knew I would have rejected it. It's not happy here, and to tell you frankly, there is nothing on earth I can buy that will make me happy. What can we buy? A shiny new phone, a car, a bigger house, a yacht? LOL. This money system made us aspire for mediocrity instead of greatness. I don't know but I am pretty sure if man did things for art and science (and not for money) then I am sure we'd all be much much happier. Think inventing things or curing the sick because you absolutely love it not because you'll get paid for it. Our personal goals are not deep anymore if you ask me. The "modern" life has been mediocre due to the money system. Life lacks  depth and soul. Even the rich guys couldn't truly live. For most  people, all they care about is how to make more money. People now only  get education so they can get jobs instead of actually learning  something new. Then people get trapped in the rat race and eventually  dies trying to get out of it. We get old very fast, we forget to live. Now since society got used to this disgusting way of life, we pretty much have no choice but to deal with it until of course we all agree that this shit is pathetic and do something about it. So YES, all children (even adults) should get all the basic necessities required to live a quality life. Because if this is the case then we can actually spend our short life on things that makes more sense than working for a paycheck. And I don't mean bumming around, I know you always wanted to do MANY things other than working 9-5 but never have the TIME to. Again, man should live for absolutely free of charge like nature intended so that we can actually live and aspire for greater things instead of working for pieces of paper that can't buy life no matter how many pieces of it you have.

The counter argument is that if you feel your existence is undesired the remedy is in your hands.

One important point from this parable is about the responsibility of parenting. If a couple is not confident that they can provide a minimum of care to their child, then they should abstain from procreating. A great many people reproduce when they should bloody not.

Since the nature did not ask us to be born at all, it entitled us to few things, tolatate our abusing it and certain expiry date to limit our abusing it. Coming back to your question, your professor missed mentioning free access to billion dollars money to each of us and eternal life. And since parents did not ask a particular child to be born and are providing those all free things, they are also entitled to few privileges like getting an intelligent, obedient and disciplined kid. Free education, health and housing - housing your parents may make sure that you get housing till your death. But how health and education fit in that list? In most of the developing countries Parents, whoever could afford, do take care of their child's education and health. And as far as I know most of the developed countries offer free education till certain level and free healthcare too. These may be referred as free but you or someone somewhere need to pay taxes for these facilities to be provided. If no one pays taxes no government will provide those free things for long. For all these to be provided as free Government could ask parents to get permission for conceiving a child. Free is an ambiguous term, nothing comes truly free in this world.