In my opinion there is no advantage at all to squatting in running shoes.They are designed to absorb shock and cushion the foot against impact.
From a back squatting point of view the last thing we need to be doing is compressing the shoe instead of driving up.
Robbing us of that vital power at the bottom. It’s also not in their design brief to handle sideways pressure put on them in the squat.
Disadvantages ALL OF THE ABOVE
Flat soled shoes(Chuck Taylors and the like)
Price – Although the Chucks aren’t that cheap the copies start at around $20
The main advantage is very little compression of the sole making the drive out of the bottom of the squat more direct.
It can be a little more difficult to get the required squat depth, this depends on a lot of factors but flexibility is the main one. There can be grip issues as well mainly because they’ve be designed as court shoes and the hardness of the rubber sole reflects that. Its a good place to start for very little money if you’re new to the sport.
Flat soled molded shoes(deadlift shoes) Price Starting at around $60 to $180
The same as the flat soled shoes but with the added advantage of a softer rubber sole and with far more grip. Most of these shoes have been designed with the deadlift in mind.The soles are thinner than the Chuck Taylor types and more flexible.
Again the same as the Chuck Taylor types with a slightly thinner sole that could have the knock on effect on getting Squat depth. The price is also a lot higher
Weightlifting / Squat shoes. Price.Starting at around a $115 and going on to $450
As with the flat and molded sole shoes, the solid rubber, composite or wooden soles offer excellent drive out of the bottom of the Squat with the added advantage of making squat depth easier.The higher heel normally between 20mm and 45mm(IPF rules have max of 50mm) It comes down to personal preference as to how high the heel is.
Generally there are a lot of factors involved here but is easier to get depth with a higher heel.
Heel height and the length of the foot is also a factor, the shorter the foot the more the tilt and at some point the higher heel becomes counter productive. Again everybody is different and some things work for people that probably shouldn’t.
Mainly cost but no other disadvantages.