Can a landlord make a tenant pay the utilities for an entire rental when the landlord is renting out an illegal apartment that is on said utilities? And if the tenant refuses to do so can they be evicted?

What is the written lease agreement ( or is there one)

If there is no agreement, and you are on month to month, basically you can be evicted with a months notice.

Was this other apartment there when you moved in ? Where you aware of it ? Is the rent you pay set low enough to compensate for this other apartment ?

But you can just report the legal apartment, and code enforcement may close either it or both your and the other apartment. ** you are assuming your apartment is legal,

If you can not work out arrangement with landlord, for percentage for electric, or some other payment, your best option is just to move.

Can a landlord make a tenant pay the utilities for an entire rental when the landlord is renting out an illegal apartment that is on said utilities? - No

And if the tenant refuses to do so can they be evicted? - In effect, because reporting the landlord doesn't mean you get to live there paying no utilities. He will be ordered to comply with codes and landlord laws, and often the illegal apartment can't meet code, and the whole operation shuts down.
If you like it and don't really want to move, negotiate. But it takes a lot of skill to politely threaten a landlord with being reported if he doesn't install separate meters, or cover it all himself. Although he can just respond by raising the rent if you are month to month.