
The answer is possibly but maybe not for the reasons you think.
Her's the sales pitch from the folks that produce and sell this stuff: it's a pure, natural antiseptic anti-fungal solution that has germicidal tendencies and is most often used as a first aid tool.
Now, the warning you should know: Different people react differently to this product and because skin sensitivity various it's important to only try this out on a small patch of skin before regular usage. Also, like most similar solutions please keep it away from your eyes and away from children.
In terms of how this is helpful for ingrown toenails it won't magically heel up your toenail in such a way that the problem doesn't persist and it won't immediately alleviate pain but it should help serve to greatly reduce the risk of any infection.
So from a practical standpoint you should have your bases substantially better covered if you use a germicide solution like this in conjunction with something like Outgro.