If my employer merges with another company, can I be required to sign a non-compete as a condition of continued employment?

No, unless the employer gives you a written copy of the non-compete agreement, and at least 7 calendar days to consider whether to sign it. Public Act No. 13-309 -a new law which becomes effective on October 1st- created those two requirements. However, an employee can waive her rights to receive a written copy of the non-compete and have at least 7 calendar days to decide whether to sign the non-compete if she signs a separate document waiving such rights.