I've got an older (classic?) black plastic MacBook that can't be upgraded past OS X Lion (10.7.5). It looks like it's no longer receiving security updates, and critical software (like web browsers) have also discontinued support. This makes me nervous to use it on today's internet. Should I install something like Windows 10 or Linux Mint instead? (If it weren't for security issues I'd be fine keeping it running as-is.)
Yes, I would definitely recommend installing an operating system which is currently supported with security updates. Whether that system is a version of Windows, Linux or something else is entirely up to your likings.
Running an old, non-supported OS X on a machine connected to the Internet is not a good idea, if you have problems with it being compromised.