Bread and butter pudding - but with juice?

Yes, you can use fruit juice instead of some or all of the milk in bread pudding. As Juhasz points out, whether or not the result will "really" be bread pudding is open to debate.

Since bread pudding is basically bread + custard, I looked up recipes for custard made with fruit juice and no milk. There are some custards made with fruit juice substituted for some of the milk. For example, Soft Orange Custard is butter, sugar, eggs, flour, salt, orange juice, lemon juice, orange zest and milk.

I found one juice-only custard in Joy of Cooking by Rombauer, Becker & Becker, 2006 edition. The recipe is Lemon-Orange Custard Filling; ingredients are lemon & orange zest, lemon & orange juice, water, sugar, flour, salt, and egg yolks or eggs. Most other custard-like fillings or sauces made with only fruit juice are thickened with cornstarch, like a pudding. Curd (lemon, lime or orange curd) is a custard-like filling or spread, made with citrus juice and no milk.

The custard in bread pudding is usually thickened with eggs, but there are some recipes that use cornstarch. Fruit juice might substitute well into one of those recipes. There are even recipes that don't use any kind of thickener. The basic idea of bread pudding is quite simple, and fairly flexible.

I substituted apple cider for the milk in my regular bread pudding recipe. I also reduced the sugar to compensate for the sweetness of the cider, and omitted vanilla extract and nutmeg. It was delicious. The flavor of the cider worked well with the flavors of cinnamon and raisin. But, the texture was not quite right. My guess is that the missing elements are some protein and fat from the milk, and that could be made up by adding more eggs and butter. I will definitely make this again, with some modifications. Here's the recipe I made, with suggested modifications at the end; the original is from Joy of Cooking.

Apple Cider Bread Pudding
------------------------

16 oz sliced bread, stale but not hard (I used fresh bread, lightly toasted)
3/4 cup raisins
4 eggs
3 cups apple cider (the original recipe calls for whole milk)
1/4 cup sugar (the original recipe calls for 3/4 cup sugar)
3/4 tsp ground cinnamon
pinch of salt
butter

Butter a 2-quart baking dish. Cut the bread into 1/2-inch cubes. 
Spread the bread and raisins in the baking dish. 
Whisk together the other ingredients and pour over the bread. 
Let sit until liquid is mostly absorbed, 10-20 minutes. 
Place the baking dish in a hot water bath, and bake at 350 F for about an hour.

Suggested modifications: Add a couple of extra eggs and some melted butter to
compensate for the missing fat and creaminess that the milk would have provided. 
Increase the cinnamon to 1 tsp, or even 1-1/2 tsp. 
Possibly add back the 1 tsp vanilla and/or the 1/4 tsp grated nutmeg from the original recipe.

There are possibly two questions hidden in the original question:

1. can I make a recipe that's like bread pudding but without eggs and milk?
2. Would this be bread pudding?

The second question can't be answered. You can call a dish whatever you want. I would think that most people would expect bread pudding to involve custard and bread, but it's easy to imagine a cheeky molecular gastronomist making some dish that is thematically similar, but contains none of the expected ingredients.

As to the first question, csk has already added a comment with an eggless "bread pudding" recipe. It does have milk, but one could easily imagine substituting another liquid. The typical milk substitutes - grain or nut milks - would be the most obvious choices.

Even more generally, the dish you're proposing sounds to me like a bread soup. Popular bread soups include gazpacho, ribollita, acquacotta, pappa al pomodoro, pandana. If you follow that last Wikipedia link, you'll see a claim that pandana can be made with "sugar, Zante currants, nutmeg, and so on." Such a recipe (which isn't actually linked on the Wikipedia page) may have contained milk, but not necessarily.

Incidentally, if I were going to try to make something like bread pudding, but without milk and eggs, I'd probably try applesauce and/or banana.