Tips for patriotic vanilla ice cream
You will have to find a way to separate the batches of ice cream, or you'll end up with the uniform purple, as you mentioned.

The proper way is to prepare three batches, using different food coloring in two of them. They should come out of the machine with a soft-serve consistency.  Hold the first batches in a cool environment while preparing the last ones, but do not allow them to reach -20°C or they will harden. Then layer them in a tub and freeze.

The proper approach is difficult for home cooks: the odds are that your ice cream machine uses a coolant-filled tub, which requires 24 hours in the freezer between batches. So it is only possible with careful planning and partial thawing of hardened batches, and it won't be ready for tomorrow. Unless you are one of the happy few with a compressor based machine, you'll need something else. 

There are a couple tricks you can use to prepare three batches without a compressor machine:


Split one batch into thirds, and churn each for a reduced time.  It will take more than 1/3 the time, but less than the full time. 
Add xanthan gum (or another stabilizer) to reduce the size of ice crystals, giving a softer, creamier result even when the ice cream isn't quite as frozen
Prepare white first, then red, then blue.  To keep the ice cream machine cold, you can just clean with a spatula after the red; a hint of red in the blue will only impart a slight purple tinge.  
For one or more of the colors, use one of the methods to prepare ice cream without a machine.  The linked question explains more fully.
Reserve part of the cream, whip it fully, and gently fold into the ice cream base.  Freeze in a tub, without the machine. This is best combined with stabilizers, small amounts of alcohol, or a propylene glycol based vanilla aroma. 


For a spotted look, you can use a squeeze bottle or dropper to apply dots red and blue food coloring to thin layers of white ice cream.  After each set of spots, apply another layer of ice cream and repeat.  I can't predict how far both colors will seep into the ice cream, and what will happen to the spots; if the coloring is propylene glycol based, they may stay liquid, but they could freeze hard if the coloring is water-based.  You may get purple if two drops land close together. 

To create a beautiful marbled appearance, use uncolored vanilla ice cream and red/blue colored syrups.  First prepare a syrup heated to the softball stage, and then thin slightly with water.  The supersaturated sugar solution will crystallize, but won't freeze into hard ice.  Set up the ice cream with layers of ice cream, red syrup, ice cream, blue syrup, then repeat. Stir gently for the marbled look, being careful not to over-mix or it will become uniformly purple.

Edit In the meantime, I tried this idea and noticed that when you pour the hygroscopic syrup over the ice cream, the white cream blooms into the syrup, and the syrup is also liquid enough to flow through and around the ice cream. I had partial success in creating a barrier by using a very thin layer of watered gelatine and waiting to freeze before applying the syrup. I then cut the ice into strips and turned them sideways, then scooped through the strips. The problem was that the gelatine water froze solid, resulting in unpleasant brittle water plates. I think that using a non-saturated syrup and gelling it with gelatine will help, but haven't tried this yet. Xanthan and guar can't thicken the syrup, as they don't hydrate in the saturated solution. Conclusion: can be beautiful, but hard to get right. 

If you accept less homogeneous colors, you can add colored ingredients to a finished vanilla ice cream. Colored marshmallows have a vanilla flavor which should mesh well. Then you can consider making colored granitas and layering them with the ice cream.   You can color pieces of paler nuts, such as coconut, or cashews, or blanched almonds, or blanched hazelnuts.  If you use nuts, mix them in after they have dried, or sprinkle them on as a topping.  Alternately, make a colored white chocolate ice cream: prepare a softish ganache with xanthan, and freeze it in a thick, dense layer on a pre-chilled porcelain plate. Alcohol well keep it softer.  Chop it into pieces and mix with white ice cream. 

These are the ideas I can think of off the top of my head. There are probably alternatives, but these offer some excellent starting points. I leave it to you (or your wife) which are most appropriate for your skills. I've tried to indicate the most challenging approaches, but all will present some difficulties. In each case, you may get color bleeding, poor consistency from improper freezing, or difficulty handling semi-frozen ice cream. The latter presents a particularly thorny obstacle, as thawing will ruin it, but it can become too hard to work with.

If you decide to use one of these ideas, but can't work out the details, drop a comment! 

Edit see some of the ideas here implemented on our blog.