Lee-Ann:

Trying to grow strawberry runners trained to netting does not sound like a practical idea. The trained runners might produce a berry or two, but they produce best when the runners are allowed to sink roots into the ground. They cannot be supplied with adequate nutrients by the runner stem alone, and will be spindly and nonproductive.

Don Yellman, Great Falls, VA

I agree with jellyman. Strawberries would not grow well if you tried to train them like that, as the runners produce daughter plants, which definitely need to root to produce any quantity of berries.

I also wouldn't crowd them too much. Strawberries need to spread somewhat.

The runners of strawberries are not like the runners of ivy ... or other plants whose life cycle includes attaching to something to achieve vertical growth. They don't make those curling tendrils to help them .... they want to root into the soil.

Thank you for all the information about growing strawberries. You are giving me much needed information for this new project.
Let's switch to raspberries...vertically, or not?
Thanks,
Lee-Ann

Raspberries grow vertically, but I don't think they're all that well suited for square foot gardening. They tend to send up new canes all over the place, so you'll have trouble keeping them contained in a small area. Also, you don't want to plant them up against a fence, as you'll only be able to pick half of the fruit. Leaving fruit on the plants to rot can encourage fruit rots.

Raspberries also need to be grown on a simple trellis. I don't think that the netting you describe would really work as a trellis.