North America isn't one of the top destinations for working during your gap year, but that doesn't mean it isn't possible. Opportunities usually involve some planning ahead, as you're unlikely to simply turn up somewhere and find a job, and you may well need to arrange the appropriate visa.
Volunteering opporunities aren't quite as limited, and they're not too difficult to find if you know where to look, especially in the Caribbean.
Unless you've arranged otherwise, the only temporary job you're likely to get in the USA is working as a summer camp counsellor. Lots of travellers do this every year, giving them the opportunity to sleep in the great American outdoors, meet the locals, and gain valuable experience. The visas for camp workers will usually allow you to remain in the US for 30 days after the job has finished to travel.
Most of the jobs available in Canada are based around the tourist industry, such as working at ski resorts during the season. Many of these will take on temporary workers, but again, it's usually necessary to arrange this before travelling.
There's a higher chance of finding paid work in Mexico, as some farmers may take on extra hands at certain times of year. The work will be hard and the pay won't be good, but it's better than nothing! Ask around the locals to see if any opportunities are going.
As in most places of the world, the bulk of volunteer work in North America is wildlife care and conservation and working with children. The USA and Canada have numerous wildlife sanctuaries and national parks that will often take on temporary volunteers, giving you the opportunity to work with some of the most amazing animals on the planet. Volunteering on an American ranch is guaranteed to make you feel like a real cowboy!
There's also plenty of conservation work in the Caribbean, the exotic marine life there in need of constant care and protection. The Caribbean is also where you'll find the childcare volunteer positions, as the less-wealthy islands often have a need for people to work in schools, orphanages, and other such places.
As with any volunteering project anywhere in the world you should think carefully about your natural interests, strengths, and weaknesses before committing to anything. These placements are usually a lot of hard work - part of what can make them such a great experience.