Keeping aside the dual citizenship, while moving to Canada my wife wanted to continue to work for her employer and the same team. According to her HR she couldn’t be on the US payroll (because here visa status wouldn’t be the same; this might not apply to you) after moving to Canada. The plan was for her US employment to terminate and for her to be hired by the Canadian subsidiary on the next day, the internal systems would still keep tenure etc.
There are many reasons why your employer won’t be keen to keep you on the US payroll and there are also a few reasons that it’s not in your interest either. Some of these have been discussed in other threads on this forum, so I will not delve into details but just lay out some of the bullet points.
From an employers perspective, the main issue is that they wouldn’t be compliant with Canadian Employment laws and standards. They wouldn’t be making your CPP contributions and would be liable to a heavy penalty. Also the employment contracts between US and Canada are quite different with Canada having significantly more employee friendly terms when it comes to termination of employment. So your employer may not be able to part with you at-will.
From your perspective, the main thing to understand is that Employment Insurance (EI) in the US is managed by individual states, so depending on the state of your residence you could be eligible for a particular amount and duration. In Canada EI is federally managed and if you’ve not made contributions as a part of your paycheck you’re not eligible. Since you’re not a resident of any US state and have not made contributions in Canada; you’re screwed.
Since your company already has a presence in Canada, I think the most straightforward path is for your US employment to be terminated and for you to be hired by the Canadian subsidiary. Being a US citizen you are still responsible for paying and filing taxes with IRS irrespective of your country of residence. There is a tax treaty between US and Canada, so you will avoid double taxation. But in simple terms on your income x, you pay max(Canadian Taxes(x), US Taxes(x)). In practice Canada has higher taxes than US on most moderate to higher income brackets so you won’t have anything to pay to Uncle Sam, but you should do your own due diligence or consult a professional when it comes to taxes.