Re: K1 After Entry Denied
No easy answer. They found out that you were working illegally in the U.S. - the one day overstay isn't such a big deal of itself, beyond disqualifying you for visa waiver, but the unauthorized employment could very much complicate your future immigration efforts. The body of your post doesn't mention a visa denial, but the subject does - if you've already been denied the K1, I suggest consulting an immigration lawyer about your options. I assume an INA or CFR number was annotated in your passport when you were denied entry - can you share that information?
If your fiancee were to obtain a K1 for you, you would have to marry within 90 days of your entry into the United States. There's no version of a K1 that doesn't involve marriage.
Re: K1 After Entry Denied
Because of the issues you have had, I would recommend an Immigration Attorney.
Please also see my post about K-1 Visa's. I went through the process bringing my guy over from Canada, and described the process.
With a K-1, you have 90 days to marry, from the day you enter the U.S. You then have to go to the USCIS office and show a copy of the marriage document, set up an appointment for fingerprinting, and an appointment for an interview.