Why is dichloromethane used to extract the eugenol in a distillation? And why is it used
Instead of hexane or benzene?

Are there any disadvantages to using dichloromethane as a solvent?

The only reasons I can think of is that 1) eugenol is more soluble in dichloromethane than it is in hexane or benzene or 2) eugenol is more soluble in dichloromethane and other, undesired substances are less soluble in it.

Dichloromethane, and all chlorinated hydrocarbons are relatively toxic and hard to dispose of. Destruction requires fairly high temperatures in an incinerator. Of course, benzene is very toxic also -- and a suspected carcinogen to boot. Hexane is much less toxic than either of the other two.