Canada

I have a little bit of good news for you, and lots of bad. The good news is that as soon as someone becomes an asylum or refugee claimant in Canada, they can become eligible for IFHP which covers prescription drugs, see https://www.canada.ca/en/immigration-refugees-citizenship/services/refugees/help-within-canada/health-care/interim-federal-health-program/coverage-summary.html

(See also a recent example of an American citizen who got IFHP after applying for aslyum at the Candian-US border, https://www.tiktok.com/@katie/_katie2113/photo/7478861387322625322 )

And now the bad:

If the refugee claim fails, it's very tough to get into Canada afterwards. See https://vancouverimmigrationblog.com/is-it-possible-to-go-from-a-failed-refugee-claimant-to-an-economic-immigrant/ and https://www.canadavisa.com/canada-immigration-discussion-board/threads/trv-for-previously-refused-refugee-claimant.123089/#post-2004890

If you did get approved for asylum or as a refugee, then you need to be careful never to return to the US or use or renew your US passport (or similar like passport card I guess), at least not until you've taken the oath and become a Canadian citizen.

If you do either, you risk losing it all and getting deported back to the US: https://www.canadavisa.com/canada-immigration-discussion-board/threads/refugee-status-cessation-and-prs-applying-for-citizenship.333455/

Finally, I've been assuming that you're a US citizen. If you are not - most folks who make a refugee or asylum claim at the US-Canada border are denied because of the Safe Third Country Agreement (STCA) between Canada and the US, and so will usually be summarily returned to the US. (Somehow US citizens don't fall under this agreement.) So don't try this if you're not a US citizen, unless you're able to consult with a lawyer first.

So if you have any other way to get to Canada (you qualify for PR / Express Entry directly somehow and have enough points to get in - possibly because you are fluent in both French & English and have a masters; or you know someone who can get you a job offer that qualifies for a work permit under NAFTA; or you have enough funds to apply as an international student; or you are under 35 years and can qualify for a working holiday visa) - try that first. The consequences of failure there are less severe.