If a company eliminates your position and offers you an unrelated job for the same pay, can you collect unemployment insurance if you quit?

You can file and hope you qualify but the company will point out that you have had an opportunity to continue to work and as such, you have quit voluntarily, which will disqualify you. One of the eligibility requirements is that you accept suitable work - refusing to accept suitable work makes you ineligible. If the job you get offered is at teh same pay grade and you can do the job, then that is suitable work. Also, any company can transfer you at any time to any other job - refusal to accept the job can lead to termination for cause.

The co-worker's job was being eliminated. She was not given an "ultimatum". She was given a choice = stay and do a new job or leave.

Will you get unemployment insurance if this happens to you? There is probably not an across the board answer to your question:

What does "unrelated job" mean?? Was the co-worker previously a nuclear scientist and was offered an administrative support position. Yes, she would probably be granted unemployment insurance. Or was she doing an administrative support role in one division and offered a transfer to a different kind of administrative support role? She probably would not be eligible for unemployment.

no, you won;t get unemployment (which is maximum 50% of your old pay and only paid for 26 wks), and is taxable income - and if you are getting paid the same, why would you quit?

No.