Dear Canadian Citizen,
I am attaching below a link for your reference. This is official announcement which has the higher authority than what I can tell you.
Unless you have a work-related purpose or immigration purpose, supposedly, you cannot apply for a social security number. Once you have a work related visa, you can apply for social security number. But I am not saying it is definitely no. You can and should always ask the Social Security Administration or retain an attorney to inquire further.
If your purpose is to file tax return, open bank accounts, etc., you can apply for a tax identification number with the IRS for purposes like these.
Please feel free to follow up with your questions, otherwise, I am ready for you to evaluate your satisfaction to my answer.
Fiona Chen, MPA, Ph.D., CPA, ABV, CFF, CITP