Can my 19 year old full time college student claim himself this year to be eligible for the first two stimulus checks

[Hal_Al]

No.

Under the CARES Act, if you (the student) are claimed, or qualify to be claimed, as a dependent on someone else’s 2019 return you cannot receive a stimulus check, in 2020.   If you qualified as a dependent for 2019, but will not be for 2020, you will most likely get it in 2021, when you file a 2020 tax return.

Note that the requirement is not just whether you are actually claimed as a dependent, it's whether you qualify to be claimed as a dependent.