Thanks so much for the note and for seeking out other threads. I did reach out to both Square and GiveWP, and GiveWP gave me a workaround by using the Square API directly, which does not use a token and does not attempt to re-fire every week and should be stable. If anyone has the same issue, here is the note from GiveWP and the link for the API workaround. I followed the instructions and it seems to be working fine now.
From GiveWP:
Give Square sets a wp-cron that is supposed to fire weekly to renew the token at Square. If something is blocking wp-cron, it can cause that (the Square plugin to quit working). A more secure way to connect is to use the API key method. Our doc on connecting to Square has a section called "Connecting to Square with the API". Please go ahead and disconnect your current connection, then set it up again according to those specific instructions:
https://givewp.com/documentation/add-ons/square-gateway/
