Gave tenant inaccurate 30 day notice- can they stay 60 more days?

[Leah C.]

I gave a Chicago tenant with a month to month lease a 30 day notice a few days after the new month started (big mistake, I know) and did not ask for them to pay other rent, just to leave in 30 days. 

I got a letter from a pro bono attorney they contacted stating that they want to avoid court and can stay in the unit for up to 60 more days because we failed to give the written notice– is this true?
Before I got this letter and spoke to tenants in person, they said the 30 day notice is invalid and appear to think they don't need to pay rent this month like a regular month....

Now I'm worried if they don't have to leave until the fall, it's another way to delay and we'll just have to start the eviction process later, which won't happen in the winter. We really need them to leave ASAP.
I'm in the process of finding a lawyer now who can help me figure out the best next step. Thank you for any suggestions.