No, do not do the COBRA thing. Opt in to your employer's group coverage. COBRA expires. You are not responsible for optimizing your employer's health insurance premiums and will probably have far less of an impact on them than you're worrying about.
Don't listen to this. Don't let yourself have any gap in coverage and if the new plan isn't as generous with the coverage then it may pay to cover the COBRA premiums and use your old insurance as the primary.
There will be a gap between quitting and getting the paperwork and mailing in your premium. They will reactivate your old insurance retroactive but that's something to be aware of.
Don't get on COBRA with a plan to stay on it, I think you're saying, and I agree.
If your new employer doesn't offer health care, go buy some: switching jobs is a "qualifying life event" and lets you buy outside the yearly sign-up window.
If they do offer health care, then you're set. Wait for that. This one-week gap isn't going to mean anything or change anything.
Yeah, I could have been clearer about this. At 18 employees, his new employer surely offers group coverage (if they didn't, he shouldn't work there regardless of whether he's dealing with an illness). And of course, if temporary COBRA coverage is the only way to ensure he has no coverage gap, he should make use of COBRA as well.
Just don't do COBRA as an alternative to the employer's group plan.