Examples in there of employee healthcare woes... sure with we just had universal healthcare.
The typical flow would be: current insurance continues unaltered for the rest of the month. Following month you're automatically covered under same policy through COBRA provisions even if you take no action. (Obviously some employers would include multiple months of employer-paid coverage, but this example is a worst-case scenario if they don't give any additional coverage in severance).
Somewhere in the 60 days following getting laid off, you need to decide between A) sign up to continue your existing insurance plan through COBRA B) sign up for a new insurance plan through healthcare.gov (subsidized if low income) or C) sign up on your spouse's plan if applicable.
It's a hassle, but you have a couple months to figure out what's best for you, and you can continue any ongoing things without interruption.