Definitely don't use a box colour because it will grab onto the hair and be more difficult to remove later on, and once you get into that cycle it is difficult to break out of it.
Although you may not want to, we would always recommend going back to the salon that did the colour service because they know what product was used and would therefore be best to change it for you.
When colour is removed from the hair like in a balayage service, the resulting tone can be a little warm or golden, based on the amount of natural red pigment in the hair itself. To counteract the gold tones, an ashy toner is used and this is probably what has given you the grey colour, in that the hair has gone a little more ashen than you wanted.
Toners usually fade with washing because they're semi-permanent and so it should be easy to resolve. You can either purchase a clarifying shampoo, like Bumble&bumble Sunday Shampoo, which will gently cleanse the hair of some of the ashy colour, or you can request that the salon where you coloured your hair does that service for you. If you like the resulting shade, then fantastic but if not, it would then be possible to reapply a new toner over the top.
Whichever option you go for, be sure to use the clarifying shampoo on just the ends you want to remove colour from - if you had your roots coloured too and use it on them then you might remove some of the colour from the root too which you don't want. Obviously if your root colour is natural then this won't be an issue. Always be sure to use a moisturising conditioner to the pre-lightened hair in order to prevent it becoming dry and brittle.
I hope this helps and good luck,
Sophie, SWH Team Stylist & Colourist, Simon Webster Hair, Brighton