FAA has a monopoly on hiring air traffic controllers
Yes, and Pentagon has a monopoly power to hire soldiers. It is a governmental organization and any government is a monopoly by definition (which is a good reason to keep its responsibilities to a minimum, but that's another story).
Unions exist because a single employee does not have bargaining power against a corporation.
Which corporation were the air-traffic controllers bargaining with, when Reagan crushed them? Hint: public employees (be they air controllers or policemen) aren't struggling against any corporations — their employers are the taxpayers. They should not be allowed to unionize — and certainly, not strike:
strike of public employees manifests nothing less than an intent on their part to obstruct the operations of government until their demands are satisfied. Such action looking toward the paralysis of government by those who have sworn to support it is unthinkable and intolerable
— Franklin Delano Roosevelt
Without them, we'd be working 10 hours a day, 6 days a week with no benefits.
Really? So, if we get the current abysmal union-membership to, say, above 80%, we'll only have to work one day a week? For 2 hours? Wouldn't that be great!!
People aren't the same as products.
People — workers — choose to sell their labor on the free market to the willing buyers. Any attempts to make that market not free should be met with the same energetic response Standard Oil and AT&T have encountered, when they tried to become a monopoly.
They have basic needs and human rights that we prefer them to have.
Any smart employer addresses basic needs of the workers — in order to keep them happy and thus more productive. No employer is allowed to violate human rights — unions or not...