Cannot install via Composer using either dev or alpha versions.

Can't reproduce this, I don't see how that's an issue with this module.

composer create-project drupal/recommended-project my-project
cd my-project
composer require drupal/ultimate_cron:^2@alpha

Works fine.

And requiring it with just "composer require drupal/ultimate_cron" gives me a specific error:

Could not find a version of package drupal/ultimate_cron matching your minimum-stability (stable). Require it with an explicit version constraint allowing its desired stability.

Make sure you are using a recent composer version.

Closing. I found a work-around to the issue. My apologies.