I worked at a place for less than two weeks. During my employment, my employer didn't collect or ask for any personal information such as my S.I.N. birth date or mailing address. After picking up my last paycheck, my former employer asked for this information for tax purposes. I would rather not give out this information. Am I legally entitled to give out my information after I stopped working?

They have to submit a W-2 or a 1099 for the time they paid you. They need your SSN to do this.

They shouldn't have given you a paycheck without this, but yes, you need to give it to them, both for them and for your obligation to the government.

Why don't you want to give out this information? This is specifically what it is for in the first place.