What can I do if my previous employer refused to pay my vacation and sick days?

The main issue is what in fact was the policy? A policy does not have to be known to you personally to be enforceable in this instance; rather, the employer will need to follow whatever policy it demonstrably did in the past, or which is encapsulated in an employee's handbook, intraoffice policy memos, etc. If you can demonstrate that the 6 month and 2 week policy is made up--as it may well be, since it's too perfectly tuned to exclude you--and that in the past, the company has paid out anyone who was employed for at least 6 months, you would have a legal claim for the compensation. You can look to anything put out by your company in regards to this policy, as well as to the experience of other former employees, to try to show that under  the real policy, you should have been paid.

The above said, though, you may *still* not be able to collect, despite ostensibly having a legal claim, if you deliberately resigned without some good external reason (e.g. a family relocation) after 6 months and 1 week. That is because there are equitable doctrines that modify legal rights, to ensure fair treatment in many cases. One is the implied covenant of good faith and fair dealing, or the obligation that parties to an agreement, including arguably the employer-employee relationship, treat each other fairly. There are also fraud-based doctrines, which hold that if you misrepresented (or lied) to get someone to enter into a transaction or relationship with you, that can give the other party grounds to rescind, or not honor, that transaction or relationship.

In this case, if it can be shown that you likely intended all along to work just long enough to qualify for benefits, then quit, the employer may be able to make out a case that you committed fraud (lied to it during the hiring process) and/or violated the covenant of good faith and fair dealing. While its not clear they would prevail on these arguments, there is at least a chance they would.

Given the difficulty you might have in proving what their policy was, and the fact that you could still lose based on what could possibly be characterized as your deceitful, unethical, or fraudulent behavior in taking a job just to get benefits, then quiting practically the day you qualified (and even if this is not in fact what you did, from what you write, it could appear that way), it may not be worthwhile taking legal action to try to get these days paid out.