Interesting. I have no problem with sneakers, especially if doing anything that requires climbing. (E.g. ladder work cleaning gutters I'm clumsy in boots) If I'm working on roofs I want a rubber soled shoe or even a soft mocasin to minimize damage and maximize traction.
Jogging pants? No problem. But I want them neat and clean if we are going to a client. No ripped out crotch, torn knees. And the elastic has to work so don't have to constantly hitch them up. (Since I run a tree farm 85% of my work is at the farm.) Similarly if they wear shorts, they are neat. No cutoffs. Running shorts are ok. Lots of my guys come to work in jeans, but use a pair of running shorts as underwear, and peel down as the day warms up.
Long pants or shin guards when using a weedeater, or rototiller, depending.
Mowing? My present walk behind mowers have a rubber flap on the back of the bell. I've yet to have anything thrown through that, so if the guys want to wear sneakers, I've no problem with it.
I hire a lot of high school kids. If I get too inflexible, they go find somewhere else to work. Being able to suntan while weeding or mowing is one of the perks.
Big thing for me is being polite. This hasen't been a problem yet. I live near a farming community, and parents insist on polite kids. But one who is impolite would stand out, so it's something I talk about during orientation.
Lot of this depends on location. California is a lot more casual about skin than Carolina.