Depending on how comfortable you are with not putting weight on your toe, pretty much anything is valid. Properly done third-world squats should not be putting any real weight on your toe (less than walking, certainly). Hindu Squats, of course, are out. Pistol squats on the one leg probably ought to be fine. In terms of machines, most of them should not be directly involving your toes, or at least should be easy enough to modify to push with your heels.
The other concern, of course, may be foot safety. I know many people are leery of being barefoot or wearing sandals in a gym. Personally, I've never had an issue — and I think that most sneakers won't do much to protect your foot anyhow — but your mileage may vary.
Lastly, while the weather is still nice, consider making use of a swimming pool, at least as long as your toenail situation doesn't involve an open wound. The water will help support you. There may be some mild discomfort from the water pushing against the nail bed, but it will be pretty minor in my experience. If the toe isn't all the way off (a hangnail), you might also want to either clip it down or to tape it down with something water-resistant, because the nail flopping about in the water definitely can be painful.
On a side note, I've been where you are. I've lost the nail on my big toe at least three times in my life, and I've suffered from ingrown toenails a few times. Honestly, it's more or less a matter of a combination of learning ways to avoid putting weight on the toe and occasionally pushing through the pain.