It appears secure boot was the culprit. One of the reasons I didn't disable it initially, I couldn't easily get to BIOS on this Toshiba laptop. I've had it for 5 years, and every time I needed to get to BIOS I had to use Windows' advanced startup settings. The F2 and all other methods never worked.
I accidentally stumbled on another way to get to BIOS. While I was booting Zorin from my thumb drive, F12 allowed me to get to the boot menu screen. This screen has an option to "Enter Setup", which is BIOS. Hopefully, this will help someone.
Going back to the WiFi issue.
When I disabled secure boot, I had firmware-b43-installer installed. All other drivers were purged. Wifi worked.
So I tested a few scenarios:
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I uninstalled firmware-b43-installer, which meant that all drivers at this point were uninstalled. The wifi worked. So whatever wifi drivers Zorin uses, they worked. I actually don't think firmware-b43-installer driver was used at all. I think whatever Zorin uses internally for wifi adapters was used. Uninstalling it didn't change anything.
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When Zorin was installed, it installed bcmwl-kernel-source driver. So I installed this driver, to see if it works now since it was the default. Wifi worked. This time it does say under Additional Drivers that this driver is being used.
So it appears that all the trouble I had was stemming from the secure boot. Based on my testing, the Broadcom 43142 adapter works with the default third-party driver bcmwl-kernel-source and it works without it, with the default drivers that Zorin ships with.
The installation wizard wording suggests that a secure boot is necessary. One of the reasons I enabled it since I had it enabled on Windows. There should be a warning that enabling secure boot will break OS functionality. That recommended setting is to keep it disabled.
Hopefully, this will help anyone else having trouble with the wifi adapters.