Do you lace up your running shoes to run errands? Do you wear them to walk the dog? Do you wear them to workout classes or to lift weights?
If at all possible, don't. That's what other shoes or retired running shoes are for.
Running shoes are designed for running. That's it. They have arch support and padding for the forward propulsion of running. Just like you wouldn't wear climbing shoes to go for a run, you shouldn't wear running shoes to lift weights -- especially for things like squats.
There are better (and cheaper) cross-trainer shoes that would be better suited for the side to side movements of a workout class and the lower heel drop needed to optimize strength building in a squat movement.
When you put your running shoes through movements they weren't designed for, they begin to wear in ways they aren't designed to wear. Even the difference between the gait of walking vs. running can alter wear patterns on your shoes. This could ultimately lead to injury. It's not worth it.
Length of Life/Stretching Your Dollar
Running shoes are not cheap. Most of my running shoes are valued around $150-200 each. Do you really want to waste that money by wearing your shoes out running errands? You don't need the extra padding and features for errands. You can just lace up the pair of Chucks you've had since high school for that. Save your shoes for running. Typically, shoes last between 300-500 miles before they need to be retired from running. Errands and dog walking counts towards that.
Personally, I often use my retired running shoes for things like this. They're past their prime for running, but not completely too destroyed to just walk around.
Knowing When It's Time to Retire
When you use your shoes for things other than running, it can be harder to tell when a shoe is dead. If you only use them for running, you'll feel the loss of cushion or compromised arch support and you'll know that it's time to stop wearing them before you hurt yourself.
So if you can, wear shoes for their specific purposes and prolong the life of your running shoes.