I am a MB in Canada. If they want to claim childcare costs on their taxes, i.e. turn this into a legal arrangement, then they must register with Canada Revenue as an employer and deduct CPP, EI and taxes for your pay cheque as well as paying their employer's contribution. It is not difficult to figure this out, they can calculate the deductions using the government's Payroll Deductions Online Calculator (google it). They need to send a cheque in once a month, though I think it might even be possible to do it quarterly. They will have to issue you a T4 every year - again, it is not a difficult form to complete. If you are a low income earner, you may find when you do your tax return that you get a lot of that money that was deducted back, so the hit might not be so bad. To be clear, you are exactly right that you cannot be a nanny in someone else's home as a self employed person/contractor, they *must* be your employer. If you are salaried, you should look into payment for public holidays, you should look at something like the publicc holiday pay calculator, as you would probably be entitled.
https://www.labour.gov.on.ca/english/es/tools/php_calc.php. Also, in Ontario for example, the government says employees are entitled to two weeks (ten days) of vacation time after each 12-month vacation entitlement year, though some employers let their employees take this before the 12 month period ends. I don't think paid sick days are mandatory but I do think that if you are available to work when they go on holiday that legally they have to pay you, otherwise they have ceased to employ you or maybe technically have laid you off, which triggers a whole bunch of paperwork.
On the employer's side, they get to claim this as a deduction so there will be a clear financial benefit for them. You might want to point out that you are aware that they will be saving money via this arrangement, whereas you are concerned that it will result in you losing money. I'm not completely clear why they are trying to reduce your hourly rate by $1. It sounds like they want more hours at a lower hourly rate. That's crazy. Why should you work more for less money??? On top of that, they will be deducting tax, EI and CPP from your pay, so net take-home pay will be even lower.
To be frank, they are either naive or they are jerks. It sounds like they are not treating you fairly in this situation. You need to be clear on your rights and their obligations, and present all of this to them, preferably with print-outs of the relevant web pages to illustrate to them exactly what is required and what the financial implications are for both parties (they should do this but if you really want to keep this job it sounds like you will have to).
Good luck!