What would be the pros and cons of raising the age of retirement and eligibility?

Eligibility for what? I presume you are talking about Social Security/Medicare.

The big issue is that many people have to work jobs with lots of physical labor - say, as a cook at an elementary school. This is quite active, and after decades of doing such work, their bodies may not be in the best of shape and may have difficulties completing the task.

It makes people work longer, but costs less, since they are not paid as long. I'm only discussing Social Security, there are a whole host of other issues with Medicare.