You've finally been caught up with by the US government. While the discussion whether the US should tax by citizenship is valid, the fact of the matter is that it does. Your Canadian citizenship/residency has zero effect on your tax liability to the US government as a US citizen. Not filing your tax returns, especially if you owe unpaid taxes - is a crime.
Here's the problems you might be facing by "lying" about your US citizenship or ignoring it:
Banks and FATCA
Banks are required to either report accounts of US tax residents (=citizens) or not provide services to them. This is a requirement of the US government, and Canada (or any other country) has nothing to do with it. Banks who do not comply will be penalized heavily by the US.
As the result, banks come with these requests. You do not have to provide them your SSN. If you don't - they will not service you.
You may end up without an ability to bank in your own country just because you refuse to comply with the US government regulations (which is also your country). I know of people who got into a similar situation. In some countries banks are legally forbidden from reporting the information to the US government - and they plainly refuse service to anyone whom they suspect to be a US person. Canada is not such a country and banks are allowed to report to IRS, but if you don't comply - you may end up without any bank providing services to you.
Reporting Requirements
In addition to the FATCA issues, you face some more trouble. The US requires its citizens to report foreign accounts. There's no tax payment, just reporting, however not complying with this requirement brings very heavy penalties.
You could resolve this problem using streamlined process with no penalties, however in you case it may be too late. Part of the requirements for this process is that your failure to pay taxes would not be willful.
Providing the required information to the bank without resolving this issue may put you into the "willful" category, since you've obviously been made aware of the requirements and still chose not to comply.
You'll need to talk to a lawyer proficient in the issue and experienced with working with expats with similar problems.
Non-Compliance Penalties
Penalties for non-compliance, unless resolved through the streamlined method, are extremely steep. For example, for FBAR this could be anywhere up to $10K per account that you failed to report, for each year it was not reported. Consider you have a $50K on your account for 10 years - you may end up with $100K penalty. Just for that account. So the US government can (and in some cases does) penalize you so that you'll end up paying way more money that the money you actually have - just for not reporting your accounts, without any taxes due.
If the taxes are due (not for this accounts, but for you in general) - for FBAR this could be anywhere up to 50% of the worth of the account that you failed to report, for each year it was not reported. Consider you have the same $50K on your account for 10 years during which you neglected to report something that wasn't taxable in Canada but is in the US - you may end up with $250K penalty.
Things have happened.
Taxes owed
You can reduce the tax liability by using any of these (separately or in combination):
Foreign Earned Income Exclusion - up to about $100K (amount adjusted to inflation every year) of earned income can be excluded from your US-taxable income. That includes salary, self-employment income, etc.
Foreign Tax Credit - income you cannot exclude using (1) is taxable in the US. But if you paid taxes in Canada, you can use the taxes paid as a credit towards the US taxes. The calculation is not straight-forward, and you use form 1116 for calculating the credit. This covers any income, including portions of salary not covered by (1), investment income, rentals, pension, etc.
Tax Treaty - as a citizen, US tax treaties do not apply to you. However, some provisions do. For most cases, you can exclude government pensions, not treat US-sourced income as US-sourced if you're in the US temporary, and depending on the country - other things. You must file tax return and form 8833 if you're claiming a treaty provision. For Canada - there's a specific exclusion for RRSP. See below.
Canadian RRSP
Generally, accounts tax-sheltered in a foreign country are not tax sheltered in the US. So your contributions to RRSP are not tax sheltered in the US.
However, Canada and the US have an explicit provision allowing you to defer US taxes on RRSP until distribution, making them treated similarly to the US-counterpart - IRA.
This is now automatic, you no longer have to make an explicit election on your tax return for this. This change is new and recent, and is made especially because of people like you. So you did get a break here.
Self-Employment Tax
Americans self-employed abroad are liable for Self-Employment tax (about 15% of the SE income) unless there's a totalization agreement between the US SSA and the local social/national security agency. There's such an agreement for US and Canada.
There's a totalization agreement, so you're covered even if you're self-employed.
Self-Employment and Investments
You may not owe any taxes in the US. For most Americans living in Canada that would be the case, since the Canadian taxes are generally higher. For unearned income the US and Canadian taxes may differ, so you may have issues there.
Specifically the most problematic area is investment in PFIC. If you have any non-US mutual fund holdings - you're stuck with PFIC. Some foreign companies are PFICs even if they trade as individual stocks, depending on their portfolio. Some companies may have not been PFIC last year, and became this year, because their portfolio holdings changed.
PFICs are taxed very aggressively by the US, and the best option is to use the Mark to Market election. You can only do that if you timely filed your taxes, so if you have mutual fund investments outside your RRSP and you didn't file such an election - you do owe US taxes.
Foreign Trusts, Companies, Gifts and Inheritance
All of the above is reportable (gifts/inheritance - if exceed $100K), and in case of companies in which you're a significant shareholder/owner: additional taxes may be due.
Penalties for non-compliance are very steep (start with $10K for not filing the forms, go up if taxes are due).
Talk to a lawyer/licensed tax adviser if it is relevant to you.
Bottom Line
Providing or not providing SSN to your bank is the least of your problems right now. I suggest you find a tax attorney/adviser specializing in expats in Canada, and get your US tax situation in order and in compliance. Once you've done that, future filings won't be too expensive as you'll be filling the same forms over and over again, just changing the numbers. But you need to start doing this properly and not ignoring it.
Using professional counsel and trying to make it through the streamlined process will probably allow you to end up with least, if at all, taxes owed. If you go through OVDP (not to do without a legal counsel!) - it will cost more.
If the IRS come after you without you first starting either streamlined or OVDP process - it will cost you the most.
Taxes assessed by the US government can be collected from you in Canada using the Canadian judicial system and reciprocity agreements.
The statute of limitations starts with filing your tax return, for each year. By not filing, you keep the statute of limitations open forever in your case.
"Go Pound Sand" as the Solution
Generally, becoming a criminal hoping that the law enforcement won't be able to enforce is an ill-advised solution. While I understand your frustration, you are a US citizen, and choosing to break the law just because you don't like it will most definitely backfire.
Consider the following situations:
Your employer wants to send you for a business trip to the US as part of your job. You'll have to refuse, obviously, and you'll also have to explain that you refuse in order to avoid being arrested in the US. You'll have to explain to your employer that you're a criminal trying to avoid prosecution in the US. You'll probably mention that you disagree with the law based on ideology, but it probably won't make you any less unemployed the next day.
You want to take your kids to Disneyland. Oh, wait, you cannot. You'll have to explain to your children that their father is a criminal and they cannot see Mickey Mouse because of that.
You may want to have a connecting flight in JFK because it saves you a $1k on a plane ticket to the Disney park in Paris to which you're taking your kids because you don't want to explain to them that you're a criminal. But again - you cannot take that, since you don't want them to see you arrested.
You may find some partners and start a company because you came up with an awesome "next Facebook" idea. But alas, once investors learn that you're a US citizen avoiding your reporting requirements - they'll run away because the company is a tainted CFC and is in trouble with the IRS. You'll end up being sued by your partners and driven out of your "next Facebook" company with not compensation.
You may end up being sick of all that and go to the nearest US consulate to renounce your citizenship. Ooops, you cannot do that until you pay all your back taxes, penalties, interest, interest on penalties, and then some more.
Are you sure you want to go there, and suggest others follow?
All that assuming the US won't be able to come after you in Canada (tell that to Americans in Switzerland).
