My child turned 18 in April 2019. He is not in college, but started working full time in June of 2019. He still lives with me. Can I claim him when I file? Can he file?

You should be able to claim him as your dependent under the Qualifying Child rules if he meets all the requirement.  He can file his own tax return, but if you claim him as a dependent then make double sure that he indicates on his tax return that he can be claimed as a dependent.

To be a Qualifying Child -

1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
2. The child must be (a) under age 19 at the end of the year, (b) under age 24 at the end of the year and a full-time student or (c) any age and permanently and totally disabled.
3. The child must have lived with you for more than half of the year. Temporary absences while away at college are considered living with you.
4. The child must not have provided more than half of his or her own support for the year.
5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child.
6. The child must be a U.S. citizen or U.S., Canada or Mexico resident for some portion of the year.
7. The child must be younger than you unless disabled.