Play the original free versions on Newgrounds (if unblocked) or purchase The Henry Stickmin Collection legally. Then, politely ask your IT department if you can use it during lunch breaks. Many workplaces are reasonable if you ask permission first.
The Ultimate Guide to Playing The Henry Stickmin Collection Unblocked at Work
Some websites offer mirror versions of blocked games, allowing users to access them.
The "choose-your-own-adventure" format allows you to make a few quick choices and see the hilarious outcomes in minutes. Failing is Fun: