Not going to happen. Canada has immigration laws and it enforces them.
Americans are free to visit Canada for you to 180 days each year. Visitors may not attend school, work, or look for work, nor can they obtain driver's licenses, ID cards, social insurance numbers, health insurance, open a bank account, etc. All visits are at the discretion of Canadian Border Services. If your wife is pregnant they may ask for proof of travel medical insurance. If your wife shows up at a Canadian hospital pregnant you will be asked for insurance, billed tens of thousands of dollars, or asked to return to the US for treatment. If there is any reason to think you can't or won't pay, they'll call Canadian Border Services and have you deported. If you are found trying to work illegally, you'll be deported. Unlike the United States, it is virtually impossible to find illegal work.
Having a child born in Canada gives that child Canadian citizenship. It does absolutely nothing to allow the parents to stay. Children can not sponsor parents for permanent residency until they are at least 18 years old.
If you wish to live temporary in Canada you'd need either a temporary residency (visitors) visa, a study permit, or a work permit. A visitors visa allows you to visit Canada for up to two years. It requires showing many tens of thousands of dollars and doesn't now permit you to attend school, work, or look for work. A study permit requires that you apply for and are accepted into a post-secondary school which accepted foreign students. Most schools will run you around $20,000 per eight months in tuition, plus $30,000+ a year in living costs for three. You are not permitted to work while in Canada except certain on-campus jobs or, with a study work permit, part time jobs. When school ends, you need to leave.
In order to apply for a work permit requires that you have a job offer from a Canadian company which has permission to hire foreign workers. Permission requires the company to obtain a labour market opinion (LMO) report which show a shortage of Canadians in that profession. This is limited to either highly skilled professions (doctors, engineers, nurses, specialized trades, etc.) or highly temporary jobs (such as seasonal farm workers, ski resorts, etc.) Work permits only last for a few years or until the job ends.
If you wish to stay in Canada permanently, you require permanent residency. This is extremely hard to obtain. You must have either: a highly skilled job over (as per work permit) in specific industries and then complete for one of 10,000 positions each year; immediate family in Canada to sponsor you; a net worth of $1.6M and invest $800k in Canada; graduate from a Canadian university in certain programs and find work within Canada within a year; obtain a provincial nomination (doctor willing to work in some remote community, start a $600k farm in Saskatchewan, top professional athlete, world class performing artist, etc.).
All applicants must submit medical information, background checks, and financial checks. Most require a minimum of six months living expenses ($13,000 per applicant).