I would make a couple of changes to your first example:
- Use should instead of can.
There's nothing to stop a person with a criminal record from applying for the job, so that person can apply for the job, but shouldn't because they will be automatically disqualified.
- Use someone/somebody instead of anyone/anybody.
This is debatable, but since the unspecified person has a criminal record, it's more specific than just anybody. So:
Someone with a criminal record should not apply for this job.
Your second example is fine. I personally would say "Anyone," but you could use anybody, anyone, somebody or someone.