Can I use carrot juce (from juicing carrots) in place of the water in a carrot recipe? I am just using a box mix.

Yes, absolutely! There are plenty of recipes on here with just that kind of substitution! If you were using something substantially sweeter instead of water (say, soda, or margarita mix or something along those lines), you could reduce the amount of sugar in your recipe (supposing you're doctoring your box mix). Otherwise, though, just use your juice. It'll be delicious! HTH!

Ditto to Emiyeric :0)