Looks like boxwood. I would say dig out with a nice root ball, trim the plant back so as to open it up for more light and air and plant them in the ground again. Next year you can put them in a training box. Usually major root pruning and repotting should be done in spring around March April time frame.

Nice starting material I must say.

Hey, thanks for the response!

I have to get these out of the yard now and can't put them back into the ground anywhere. Would there be any point to digging them out and putting them into large containers? If so, I'll also need to go ahead and reduce the size of them or I won't have anywhere to keep them. I feel like I would have a hard time killing these things if I tried. They are in hard clay soil and only get rain water.

If this is pointless, they'll just have to be discarded. I would love to save at least a couple of them to get my feet wet with bonsai, but I also don't have years invested in growing them for that purpose, so scrapping them is acceptable to me.

By the way, I live in the Dallas area, so it's hot and humid for the next few months.

In Dallas, you should no problem keeping them in containers outside all year round. And happily experiment away. Reduce the upper growth to as much you think can manage. I would may be prune some a lot and some other less to see how they respond next year. I would also aim for at least foot wide root ball. If you can manage a larger root ball I would even try cutting say three wedges in the root ball and put some free-draining mix. Then pot them up with free draining mix, water very well and keep in shade for the next 2-3 weeks.

I wouldn't worry about killing them. I, like you dug some up from an old hedge that was about 25 years old. I had them in pots for the past year and they are doing really well. I suggest cutting the larger branches back to allow more light and air in near the trunk area, this will allow for new shoots to grow which will be your new foliage moving forward. Boxwood are typically very slow growers when it comes to bonsai so don't rush into putting them into bonsai "soil/media" right away. Allow for it to flush some new growth next spring.

Your landscape looks good. If you must remove plants....They snatch out of the ground easily with a chain hooked to your truck. I prefer to collect bonsai material in late winter ahead of new growth. This will certainly get your success rate up when doing drastic pruning and removing soil from rootball.