Hi Kaylee Mae,
When I was severely neutropenic, I drank a lot of almond, soy, and coconut milk, and coconut water (I think the coconut water has a lot of sugar, but perhaps not so much with the milk).
Also, my hospital (Colorado Blood Cancer Center at Presbyterian St Luke's) served us all sorts of raw fruits and vegetables, even when my ANC was 0 during treatment. That may sound crazy, but I'm confident they are an excellent hospital and know what they are doing. The crucial part is making sure the raw fruits and vegetables are very clean with something made specifically for the task. My hospital used something EPA-approved called Victory Wash, described in the link below:
http://www.foodservice.com/food_safety/fb4.cfm
...when I got home and started slowly to be less neutropenic, I ate fruits and vegetables washed in something called NutriBiotic GSE Grapefruit Seed Extract (I still use this today). I reviewed that with one of my doctors, and he OKed this. In my parents' support group, I believe one of the patients said his doctor told him even vinegar is good to wash fresh food in when you are neutropenic (same as Marlene said).
Of course, I am not a doctor, so I'd urge you to review any of this with your boyfriend's doctors/hospital before trying. Also, I have aplastic anemia, maybe the rules are different with leukemia. But since it sounds like you are looking hard for dietary options, and eating fresh food is probably healthy if possible, I wanted to mention it.
I'm sorry to hear you guys have to go through this, but as I'm sure you know transplant can be a wonderful and permanent cure. Your boyfriend is fortunate to have you watching out for him and making sure what he eats is safe.