Universal Prekindergarten and Early Childhood Education Act of 2018 This bill authorizes through FY2024 a competitive grant program for states to establish or expand full-day prekindergarten programs that: (1) permit each three- or four-year old child in the state to be voluntarily enrolled by the child's parent, regardless of income, in the program at the public school in which the child would be enrolled for kindergarten; (2) run the length of the regular school year; and (3) are taught by teachers who possess equivalent or similar qualifications to those of teachers of other grades in the school. The federal share of such a grant shall not exceed 80% of the costs of carrying out these activities. Grant funds received under the program shall be used to supplement and not supplant other federal early childhood education funds in the state.